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Theorem subctctexmid 15491
Description: If every subcountable set is countable and Markov's principle holds, excluded middle follows. Proposition 2.6 of [BauerSwan], p. 14:4. The proof is taken from that paper. (Contributed by Jim Kingdon, 29-Nov-2023.)
Hypotheses
Ref Expression
subctctexmid.x |- ( ph -> A. x ( E. s ( s C_ om /\ E. f f : s -onto-> x ) -> E. g g : om -onto-> ( x ⊔ 1o ) ) )
subctctexmid.mk |- ( ph -> om e. Markov )
Assertion
Ref Expression
subctctexmid |- ( ph -> EXMID )
Distinct variable groups:   f, s, x   ph, g   x, g
Allowed substitution hints:   ph( x, f, s)
Proof of Theorem subctctexmid
Dummy variables y z h n w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 subctctexmid.x . . . . 5 |- ( ph -> A. x ( E. s ( s C_ om /\ E. f f : s -onto-> x ) -> E. g g : om -onto-> ( x ⊔ 1o ) ) )
2 omex 4625 . . . . . . . 8 |- om e. _V
32rabex 4173 . . . . . . 7 |- { z e. om | y = { (/) } } e. _V
43a1i 9 . . . . . 6 |- ( ph -> { z e. om | y = { (/) } } e. _V )
5 ssrab2 3264 . . . . . . 7 |- { z e. om | y = { (/) } } C_ om
6 f1oi 5538 . . . . . . . . 9 |- ( _I |` { z e. om | y = { (/) } } ) : { z e. om | y = { (/) } } -1-1-onto-> { z e. om | y = { (/) } }
7 f1ofo 5507 . . . . . . . . 9 |- ( ( _I |` { z e. om | y = { (/) } } ) : { z e. om | y = { (/) } } -1-1-onto-> { z e. om | y = { (/) } } -> ( _I |` { z e. om | y = { (/) } } ) : { z e. om | y = { (/) } } -onto-> { z e. om | y = { (/) } } )
86, 7ax-mp 5 . . . . . . . 8 |- ( _I |` { z e. om | y = { (/) } } ) : { z e. om | y = { (/) } } -onto-> { z e. om | y = { (/) } }
9 resiexg 4987 . . . . . . . . . 10 |- ( { z e. om | y = { (/) } } e. _V -> ( _I |` { z e. om | y = { (/) } } ) e. _V )
103, 9ax-mp 5 . . . . . . . . 9 |- ( _I |` { z e. om | y = { (/) } } ) e. _V
11 foeq1 5472 . . . . . . . . 9 |- ( f = ( _I |` { z e. om | y = { (/) } } ) -> ( f : { z e. om | y = { (/) } } -onto-> { z e. om | y = { (/) } } <-> ( _I |` { z e. om | y = { (/) } } ) : { z e. om | y = { (/) } } -onto-> { z e. om | y = { (/) } } ) )
1210, 11spcev 2855 . . . . . . . 8 |- ( ( _I |` { z e. om | y = { (/) } } ) : { z e. om | y = { (/) } } -onto-> { z e. om | y = { (/) } } -> E. f f : { z e. om | y = { (/) } } -onto-> { z e. om | y = { (/) } } )
138, 12ax-mp 5 . . . . . . 7 |- E. f f : { z e. om | y = { (/) } } -onto-> { z e. om | y = { (/) } }
145, 13pm3.2i 272 . . . . . 6 |- ( { z e. om | y = { (/) } } C_ om /\ E. f f : { z e. om | y = { (/) } } -onto-> { z e. om | y = { (/) } } )
15 sseq1 3202 . . . . . . . 8 |- ( s = { z e. om | y = { (/) } } -> ( s C_ om <-> { z e. om | y = { (/) } } C_ om ) )
16 foeq2 5473 . . . . . . . . 9 |- ( s = { z e. om | y = { (/) } } -> ( f : s -onto-> { z e. om | y = { (/) } } <-> f : { z e. om | y = { (/) } } -onto-> { z e. om | y = { (/) } } ) )
1716exbidv 1836 . . . . . . . 8 |- ( s = { z e. om | y = { (/) } } -> ( E. f f : s -onto-> { z e. om | y = { (/) } } <-> E. f f : { z e. om | y = { (/) } } -onto-> { z e. om | y = { (/) } } ) )
1815, 17anbi12d 473 . . . . . . 7 |- ( s = { z e. om | y = { (/) } } -> ( ( s C_ om /\ E. f f : s -onto-> { z e. om | y = { (/) } } ) <-> ( { z e. om | y = { (/) } } C_ om /\ E. f f : { z e. om | y = { (/) } } -onto-> { z e. om | y = { (/) } } ) ) )
1918spcegv 2848 . . . . . 6 |- ( { z e. om | y = { (/) } } e. _V -> ( ( { z e. om | y = { (/) } } C_ om /\ E. f f : { z e. om | y = { (/) } } -onto-> { z e. om | y = { (/) } } ) -> E. s ( s C_ om /\ E. f f : s -onto-> { z e. om | y = { (/) } } ) ) )
204, 14, 19mpisyl 1457 . . . . 5 |- ( ph -> E. s ( s C_ om /\ E. f f : s -onto-> { z e. om | y = { (/) } } ) )
21 foeq3 5474 . . . . . . . . . 10 |- ( x = { z e. om | y = { (/) } } -> ( f : s -onto-> x <-> f : s -onto-> { z e. om | y = { (/) } } ) )
2221exbidv 1836 . . . . . . . . 9 |- ( x = { z e. om | y = { (/) } } -> ( E. f f : s -onto-> x <-> E. f f : s -onto-> { z e. om | y = { (/) } } ) )
2322anbi2d 464 . . . . . . . 8 |- ( x = { z e. om | y = { (/) } } -> ( ( s C_ om /\ E. f f : s -onto-> x ) <-> ( s C_ om /\ E. f f : s -onto-> { z e. om | y = { (/) } } ) ) )
2423exbidv 1836 . . . . . . 7 |- ( x = { z e. om | y = { (/) } } -> ( E. s ( s C_ om /\ E. f f : s -onto-> x ) <-> E. s ( s C_ om /\ E. f f : s -onto-> { z e. om | y = { (/) } } ) ) )
25 djueq1 7099 . . . . . . . . 9 |- ( x = { z e. om | y = { (/) } } -> ( x ⊔ 1o ) = ( { z e. om | y = { (/) } } ⊔ 1o ) )
26 foeq3 5474 . . . . . . . . 9 |- ( ( x ⊔ 1o ) = ( { z e. om | y = { (/) } } ⊔ 1o ) -> ( g : om -onto-> ( x ⊔ 1o ) <-> g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) )
2725, 26syl 14 . . . . . . . 8 |- ( x = { z e. om | y = { (/) } } -> ( g : om -onto-> ( x ⊔ 1o ) <-> g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) )
2827exbidv 1836 . . . . . . 7 |- ( x = { z e. om | y = { (/) } } -> ( E. g g : om -onto-> ( x ⊔ 1o ) <-> E. g g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) )
2924, 28imbi12d 234 . . . . . 6 |- ( x = { z e. om | y = { (/) } } -> ( ( E. s ( s C_ om /\ E. f f : s -onto-> x ) -> E. g g : om -onto-> ( x ⊔ 1o ) ) <-> ( E. s ( s C_ om /\ E. f f : s -onto-> { z e. om | y = { (/) } } ) -> E. g g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) ) )
303, 29spcv 2854 . . . . 5 |- ( A. x ( E. s ( s C_ om /\ E. f f : s -onto-> x ) -> E. g g : om -onto-> ( x ⊔ 1o ) ) -> ( E. s ( s C_ om /\ E. f f : s -onto-> { z e. om | y = { (/) } } ) -> E. g g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) )
311, 20, 30sylc 62 . . . 4 |- ( ph -> E. g g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) )
32 fveq1 5553 . . . . . . . . . . . 12 |- ( h = ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) -> ( h ` n ) = ( ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) ` n ) )
3332eqeq1d 2202 . . . . . . . . . . 11 |- ( h = ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) -> ( ( h ` n ) = 1o <-> ( ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) ` n ) = 1o ) )
3433rexbidv 2495 . . . . . . . . . 10 |- ( h = ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) -> ( E. n e. om ( h ` n ) = 1o <-> E. n e. om ( ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) ` n ) = 1o ) )
3534notbid 668 . . . . . . . . 9 |- ( h = ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) -> ( -. E. n e. om ( h ` n ) = 1o <-> -. E. n e. om ( ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) ` n ) = 1o ) )
3635notbid 668 . . . . . . . 8 |- ( h = ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) -> ( -. -. E. n e. om ( h ` n ) = 1o <-> -. -. E. n e. om ( ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) ` n ) = 1o ) )
3736, 34imbi12d 234 . . . . . . 7 |- ( h = ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) -> ( ( -. -. E. n e. om ( h ` n ) = 1o -> E. n e. om ( h ` n ) = 1o ) <-> ( -. -. E. n e. om ( ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) ` n ) = 1o -> E. n e. om ( ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) ` n ) = 1o ) ) )
38 subctctexmid.mk . . . . . . . . 9 |- ( ph -> om e. Markov )
39 ismkvnex 7214 . . . . . . . . . 10 |- ( om e. Markov -> ( om e. Markov <-> A. h e. ( 2o ^m om ) ( -. -. E. n e. om ( h ` n ) = 1o -> E. n e. om ( h ` n ) = 1o ) ) )
4038, 39syl 14 . . . . . . . . 9 |- ( ph -> ( om e. Markov <-> A. h e. ( 2o ^m om ) ( -. -. E. n e. om ( h ` n ) = 1o -> E. n e. om ( h ` n ) = 1o ) ) )
4138, 40mpbid 147 . . . . . . . 8 |- ( ph -> A. h e. ( 2o ^m om ) ( -. -. E. n e. om ( h ` n ) = 1o -> E. n e. om ( h ` n ) = 1o ) )
4241adantr 276 . . . . . . 7 |- ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) -> A. h e. ( 2o ^m om ) ( -. -. E. n e. om ( h ` n ) = 1o -> E. n e. om ( h ` n ) = 1o ) )
43 1lt2o 6495 . . . . . . . . . . . 12 |- 1o e. 2o
4443a1i 9 . . . . . . . . . . 11 |- ( ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ n e. om ) /\ ( 1st ` ( g ` n ) ) = (/) ) -> 1o e. 2o )
45 0lt2o 6494 . . . . . . . . . . . 12 |- (/) e. 2o
4645a1i 9 . . . . . . . . . . 11 |- ( ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ n e. om ) /\ -. ( 1st ` ( g ` n ) ) = (/) ) -> (/) e. 2o )
47 simplr 528 . . . . . . . . . . . . . . 15 |- ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ n e. om ) -> g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) )
48 fof 5476 . . . . . . . . . . . . . . 15 |- ( g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) -> g : om --> ( { z e. om | y = { (/) } } ⊔ 1o ) )
4947, 48syl 14 . . . . . . . . . . . . . 14 |- ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ n e. om ) -> g : om --> ( { z e. om | y = { (/) } } ⊔ 1o ) )
50 simpr 110 . . . . . . . . . . . . . 14 |- ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ n e. om ) -> n e. om )
5149, 50ffvelcdmd 5694 . . . . . . . . . . . . 13 |- ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ n e. om ) -> ( g ` n ) e. ( { z e. om | y = { (/) } } ⊔ 1o ) )
52 eldju1st 7130 . . . . . . . . . . . . 13 |- ( ( g ` n ) e. ( { z e. om | y = { (/) } } ⊔ 1o ) -> ( ( 1st ` ( g ` n ) ) = (/) \/ ( 1st ` ( g ` n ) ) = 1o ) )
5351, 52syl 14 . . . . . . . . . . . 12 |- ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ n e. om ) -> ( ( 1st ` ( g ` n ) ) = (/) \/ ( 1st ` ( g ` n ) ) = 1o ) )
54 1n0 6485 . . . . . . . . . . . . . . . 16 |- 1o =/= (/)
5554neii 2366 . . . . . . . . . . . . . . 15 |- -. 1o = (/)
56 eqeq1 2200 . . . . . . . . . . . . . . 15 |- ( ( 1st ` ( g ` n ) ) = 1o -> ( ( 1st ` ( g ` n ) ) = (/) <-> 1o = (/) ) )
5755, 56mtbiri 676 . . . . . . . . . . . . . 14 |- ( ( 1st ` ( g ` n ) ) = 1o -> -. ( 1st ` ( g ` n ) ) = (/) )
5857orim2i 762 . . . . . . . . . . . . 13 |- ( ( ( 1st ` ( g ` n ) ) = (/) \/ ( 1st ` ( g ` n ) ) = 1o ) -> ( ( 1st ` ( g ` n ) ) = (/) \/ -. ( 1st ` ( g ` n ) ) = (/) ) )
59 df-dc 836 . . . . . . . . . . . . 13 |- (DECID ( 1st ` ( g ` n ) ) = (/) <-> ( ( 1st ` ( g ` n ) ) = (/) \/ -. ( 1st ` ( g ` n ) ) = (/) ) )
6058, 59sylibr 134 . . . . . . . . . . . 12 |- ( ( ( 1st ` ( g ` n ) ) = (/) \/ ( 1st ` ( g ` n ) ) = 1o ) -> DECID ( 1st ` ( g ` n ) ) = (/) )
6153, 60syl 14 . . . . . . . . . . 11 |- ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ n e. om ) -> DECID ( 1st ` ( g ` n ) ) = (/) )
6244, 46, 61ifcldadc 3586 . . . . . . . . . 10 |- ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ n e. om ) -> if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) e. 2o )
6362fmpttd 5713 . . . . . . . . 9 |- ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) -> ( n e. om |-> if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) ) : om --> 2o )
64 2fveq3 5559 . . . . . . . . . . . . . 14 |- ( w = n -> ( 1st ` ( g ` w ) ) = ( 1st ` ( g ` n ) ) )
6564eqeq1d 2202 . . . . . . . . . . . . 13 |- ( w = n -> ( ( 1st ` ( g ` w ) ) = (/) <-> ( 1st ` ( g ` n ) ) = (/) ) )
6665ifbid 3578 . . . . . . . . . . . 12 |- ( w = n -> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) = if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) )
67 eqcom 2195 . . . . . . . . . . . 12 |- ( w = n <-> n = w )
68 eqcom 2195 . . . . . . . . . . . 12 |- ( if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) = if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) <-> if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) = if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) )
6966, 67, 683imtr3i 200 . . . . . . . . . . 11 |- ( n = w -> if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) = if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) )
7069cbvmptv 4125 . . . . . . . . . 10 |- ( n e. om |-> if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) ) = ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) )
7170feq1i 5396 . . . . . . . . 9 |- ( ( n e. om |-> if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) ) : om --> 2o <-> ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) : om --> 2o )
7263, 71sylib 122 . . . . . . . 8 |- ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) -> ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) : om --> 2o )
73 2onn 6574 . . . . . . . . . 10 |- 2o e. om
7473elexi 2772 . . . . . . . . 9 |- 2o e. _V
7574, 2elmap 6731 . . . . . . . 8 |- ( ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) e. ( 2o ^m om ) <-> ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) : om --> 2o )
7672, 75sylibr 134 . . . . . . 7 |- ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) -> ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) e. ( 2o ^m om ) )
7737, 42, 76rspcdva 2869 . . . . . 6 |- ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) -> ( -. -. E. n e. om ( ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) ` n ) = 1o -> E. n e. om ( ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) ` n ) = 1o ) )
78 eqid 2193 . . . . . . . . . . . . 13 |- ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) = ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) )
7978, 66, 50, 62fvmptd3 5651 . . . . . . . . . . . 12 |- ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ n e. om ) -> ( ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) ` n ) = if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) )
8079eqeq1d 2202 . . . . . . . . . . 11 |- ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ n e. om ) -> ( ( ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) ` n ) = 1o <-> if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) = 1o ) )
8151adantr 276 . . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ n e. om ) /\ if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) = 1o ) -> ( g ` n ) e. ( { z e. om | y = { (/) } } ⊔ 1o ) )
82 simpr 110 . . . . . . . . . . . . . . . . 17 |- ( ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ n e. om ) /\ if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) = 1o ) -> if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) = 1o )
8382eqcomd 2199 . . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ n e. om ) /\ if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) = 1o ) -> 1o = if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) )
84 eqifdc 3592 . . . . . . . . . . . . . . . . . . 19 |- (DECID ( 1st ` ( g ` n ) ) = (/) -> ( 1o = if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) <-> ( ( ( 1st ` ( g ` n ) ) = (/) /\ 1o = 1o ) \/ ( -. ( 1st ` ( g ` n ) ) = (/) /\ 1o = (/) ) ) ) )
8561, 84syl 14 . . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ n e. om ) -> ( 1o = if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) <-> ( ( ( 1st ` ( g ` n ) ) = (/) /\ 1o = 1o ) \/ ( -. ( 1st ` ( g ` n ) ) = (/) /\ 1o = (/) ) ) ) )
86 eqid 2193 . . . . . . . . . . . . . . . . . . 19 |- 1o = 1o
87 orcom 729 . . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( 1st ` ( g ` n ) ) = (/) /\ 1o = 1o ) \/ ( -. ( 1st ` ( g ` n ) ) = (/) /\ 1o = (/) ) ) <-> ( ( -. ( 1st ` ( g ` n ) ) = (/) /\ 1o = (/) ) \/ ( ( 1st ` ( g ` n ) ) = (/) /\ 1o = 1o ) ) )
8855intnan 930 . . . . . . . . . . . . . . . . . . . . 21 |- -. ( -. ( 1st ` ( g ` n ) ) = (/) /\ 1o = (/) )
89 biorf 745 . . . . . . . . . . . . . . . . . . . . 21 |- ( -. ( -. ( 1st ` ( g ` n ) ) = (/) /\ 1o = (/) ) -> ( ( ( 1st ` ( g ` n ) ) = (/) /\ 1o = 1o ) <-> ( ( -. ( 1st ` ( g ` n ) ) = (/) /\ 1o = (/) ) \/ ( ( 1st ` ( g ` n ) ) = (/) /\ 1o = 1o ) ) ) )
9088, 89ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 |- ( ( ( 1st ` ( g ` n ) ) = (/) /\ 1o = 1o ) <-> ( ( -. ( 1st ` ( g ` n ) ) = (/) /\ 1o = (/) ) \/ ( ( 1st ` ( g ` n ) ) = (/) /\ 1o = 1o ) ) )
9187, 90bitr4i 187 . . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( 1st ` ( g ` n ) ) = (/) /\ 1o = 1o ) \/ ( -. ( 1st ` ( g ` n ) ) = (/) /\ 1o = (/) ) ) <-> ( ( 1st ` ( g ` n ) ) = (/) /\ 1o = 1o ) )
9286, 91mpbiran2 943 . . . . . . . . . . . . . . . . . 18 |- ( ( ( ( 1st ` ( g ` n ) ) = (/) /\ 1o = 1o ) \/ ( -. ( 1st ` ( g ` n ) ) = (/) /\ 1o = (/) ) ) <-> ( 1st ` ( g ` n ) ) = (/) )
9385, 92bitrdi 196 . . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ n e. om ) -> ( 1o = if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) <-> ( 1st ` ( g ` n ) ) = (/) ) )
9493adantr 276 . . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ n e. om ) /\ if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) = 1o ) -> ( 1o = if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) <-> ( 1st ` ( g ` n ) ) = (/) ) )
9583, 94mpbid 147 . . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ n e. om ) /\ if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) = 1o ) -> ( 1st ` ( g ` n ) ) = (/) )
96 eldju2ndl 7131 . . . . . . . . . . . . . . 15 |- ( ( ( g ` n ) e. ( { z e. om | y = { (/) } } ⊔ 1o ) /\ ( 1st ` ( g ` n ) ) = (/) ) -> ( 2nd ` ( g ` n ) ) e. { z e. om | y = { (/) } } )
9781, 95, 96syl2anc 411 . . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ n e. om ) /\ if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) = 1o ) -> ( 2nd ` ( g ` n ) ) e. { z e. om | y = { (/) } } )
98 biidd 172 . . . . . . . . . . . . . . 15 |- ( z = ( 2nd ` ( g ` n ) ) -> ( y = { (/) } <-> y = { (/) } ) )
9998elrab 2916 . . . . . . . . . . . . . 14 |- ( ( 2nd ` ( g ` n ) ) e. { z e. om | y = { (/) } } <-> ( ( 2nd ` ( g ` n ) ) e. om /\ y = { (/) } ) )
10097, 99sylib 122 . . . . . . . . . . . . 13 |- ( ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ n e. om ) /\ if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) = 1o ) -> ( ( 2nd ` ( g ` n ) ) e. om /\ y = { (/) } ) )
101100simprd 114 . . . . . . . . . . . 12 |- ( ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ n e. om ) /\ if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) = 1o ) -> y = { (/) } )
102101ex 115 . . . . . . . . . . 11 |- ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ n e. om ) -> ( if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) = 1o -> y = { (/) } ) )
10380, 102sylbid 150 . . . . . . . . . 10 |- ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ n e. om ) -> ( ( ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) ` n ) = 1o -> y = { (/) } ) )
104103rexlimdva 2611 . . . . . . . . 9 |- ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) -> ( E. n e. om ( ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) ` n ) = 1o -> y = { (/) } ) )
105 simplr 528 . . . . . . . . . . . 12 |- ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ y = { (/) } ) -> g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) )
106 biidd 172 . . . . . . . . . . . . . 14 |- ( z = (/) -> ( y = { (/) } <-> y = { (/) } ) )
107 peano1 4626 . . . . . . . . . . . . . . 15 |- (/) e. om
108107a1i 9 . . . . . . . . . . . . . 14 |- ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ y = { (/) } ) -> (/) e. om )
109 simpr 110 . . . . . . . . . . . . . 14 |- ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ y = { (/) } ) -> y = { (/) } )
110106, 108, 109elrabd 2918 . . . . . . . . . . . . 13 |- ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ y = { (/) } ) -> (/) e. { z e. om | y = { (/) } } )
111 djulcl 7110 . . . . . . . . . . . . 13 |- ( (/) e. { z e. om | y = { (/) } } -> (inl ` (/) ) e. ( { z e. om | y = { (/) } } ⊔ 1o ) )
112110, 111syl 14 . . . . . . . . . . . 12 |- ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ y = { (/) } ) -> (inl ` (/) ) e. ( { z e. om | y = { (/) } } ⊔ 1o ) )
113 foelrn 5795 . . . . . . . . . . . 12 |- ( ( g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) /\ (inl ` (/) ) e. ( { z e. om | y = { (/) } } ⊔ 1o ) ) -> E. n e. om (inl ` (/) ) = ( g ` n ) )
114105, 112, 113syl2anc 411 . . . . . . . . . . 11 |- ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ y = { (/) } ) -> E. n e. om (inl ` (/) ) = ( g ` n ) )
11579adantlr 477 . . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ y = { (/) } ) /\ n e. om ) -> ( ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) ` n ) = if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) )
116 fveq2 5554 . . . . . . . . . . . . . . . 16 |- ( (inl ` (/) ) = ( g ` n ) -> ( 1st ` (inl ` (/) ) ) = ( 1st ` ( g ` n ) ) )
117 1stinl 7133 . . . . . . . . . . . . . . . . 17 |- ( (/) e. om -> ( 1st ` (inl ` (/) ) ) = (/) )
118107, 117ax-mp 5 . . . . . . . . . . . . . . . 16 |- ( 1st ` (inl ` (/) ) ) = (/)
119116, 118eqtr3di 2241 . . . . . . . . . . . . . . 15 |- ( (inl ` (/) ) = ( g ` n ) -> ( 1st ` ( g ` n ) ) = (/) )
120119iftrued 3564 . . . . . . . . . . . . . 14 |- ( (inl ` (/) ) = ( g ` n ) -> if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) = 1o )
121115, 120sylan9eq 2246 . . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ y = { (/) } ) /\ n e. om ) /\ (inl ` (/) ) = ( g ` n ) ) -> ( ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) ` n ) = 1o )
122121ex 115 . . . . . . . . . . . 12 |- ( ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ y = { (/) } ) /\ n e. om ) -> ( (inl ` (/) ) = ( g ` n ) -> ( ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) ` n ) = 1o ) )
123122reximdva 2596 . . . . . . . . . . 11 |- ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ y = { (/) } ) -> ( E. n e. om (inl ` (/) ) = ( g ` n ) -> E. n e. om ( ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) ` n ) = 1o ) )
124114, 123mpd 13 . . . . . . . . . 10 |- ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ y = { (/) } ) -> E. n e. om ( ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) ` n ) = 1o )
125124ex 115 . . . . . . . . 9 |- ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) -> ( y = { (/) } -> E. n e. om ( ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) ` n ) = 1o ) )
126104, 125impbid 129 . . . . . . . 8 |- ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) -> ( E. n e. om ( ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) ` n ) = 1o <-> y = { (/) } ) )
127126notbid 668 . . . . . . 7 |- ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) -> ( -. E. n e. om ( ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) ` n ) = 1o <-> -. y = { (/) } ) )
128127notbid 668 . . . . . 6 |- ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) -> ( -. -. E. n e. om ( ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) ` n ) = 1o <-> -. -. y = { (/) } ) )
12977, 128, 1263imtr3d 202 . . . . 5 |- ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) -> ( -. -. y = { (/) } -> y = { (/) } ) )
130 df-stab 832 . . . . 5 |- (STAB y = { (/) } <-> ( -. -. y = { (/) } -> y = { (/) } ) )
131129, 130sylibr 134 . . . 4 |- ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) -> STAB y = { (/) } )
13231, 131exlimddv 1910 . . 3 |- ( ph -> STAB y = { (/) } )
133132adantr 276 . 2 |- ( ( ph /\ y C_ { (/) } ) -> STAB y = { (/) } )
134133exmid1stab 4237 1 |- ( ph -> EXMID )
Colors of variables: wff set class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709  STAB wstab 831  DECID wdc 835   A.wal 1362   = wceq 1364   E.wex 1503   e. wcel 2164   A.wral 2472   E.wrex 2473   {crab 2476   _Vcvv 2760   C_ wss 3153   (/)c0 3446   ifcif 3557   {csn 3618   |-> cmpt 4090  EXMIDwem 4223   _I cid 4319   omcom 4622   |` cres 4661   -->wf 5250   -onto->wfo 5252   -1-1-onto->wf1o 5253   ` cfv 5254  (class class class)co 5918   1stc1st 6191   2ndc2nd 6192   1oc1o 6462   2oc2o 6463   ^m cmap 6702   ⊔ cdju 7096  inlcinl 7104  Markovcmarkov 7210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620
This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-exmid 4224  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-1o 6469  df-2o 6470  df-map 6704  df-dju 7097  df-inl 7106  df-inr 7107  df-markov 7211
This theorem is referenced by: (None)
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