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Theorem subctctexmid 14034
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 4577 . . . . . . . 8 |- om e. _V
32rabex 4133 . . . . . . 7 |- { z e. om | y = { (/) } } e. _V
43a1i 9 . . . . . 6 |- ( ph -> { z e. om | y = { (/) } } e. _V )
5 ssrab2 3232 . . . . . . 7 |- { z e. om | y = { (/) } } C_ om
6 f1oi 5480 . . . . . . . . 9 |- ( _I |` { z e. om | y = { (/) } } ) : { z e. om | y = { (/) } } -1-1-onto-> { z e. om | y = { (/) } }
7 f1ofo 5449 . . . . . . . . 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 4936 . . . . . . . . . 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 5416 . . . . . . . . 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 2825 . . . . . . . 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 270 . . . . . 6 |- ( { z e. om | y = { (/) } } C_ om /\ E. f f : { z e. om | y = { (/) } } -onto-> { z e. om | y = { (/) } } )
15 sseq1 3170 . . . . . . . 8 |- ( s = { z e. om | y = { (/) } } -> ( s C_ om <-> { z e. om | y = { (/) } } C_ om ) )
16 foeq2 5417 . . . . . . . . 9 |- ( s = { z e. om | y = { (/) } } -> ( f : s -onto-> { z e. om | y = { (/) } } <-> f : { z e. om | y = { (/) } } -onto-> { z e. om | y = { (/) } } ) )
1716exbidv 1818 . . . . . . . 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 470 . . . . . . 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 2818 . . . . . 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 1439 . . . . 5 |- ( ph -> E. s ( s C_ om /\ E. f f : s -onto-> { z e. om | y = { (/) } } ) )
21 foeq3 5418 . . . . . . . . . 10 |- ( x = { z e. om | y = { (/) } } -> ( f : s -onto-> x <-> f : s -onto-> { z e. om | y = { (/) } } ) )
2221exbidv 1818 . . . . . . . . 9 |- ( x = { z e. om | y = { (/) } } -> ( E. f f : s -onto-> x <-> E. f f : s -onto-> { z e. om | y = { (/) } } ) )
2322anbi2d 461 . . . . . . . 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 1818 . . . . . . 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 7017 . . . . . . . . 9 |- ( x = { z e. om | y = { (/) } } -> ( x ⊔ 1o ) = ( { z e. om | y = { (/) } } ⊔ 1o ) )
26 foeq3 5418 . . . . . . . . 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 1818 . . . . . . 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 233 . . . . . 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 2824 . . . . 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 5495 . . . . . . . . . . . 12 |- ( h = ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) -> ( h ` n ) = ( ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) ` n ) )
3332eqeq1d 2179 . . . . . . . . . . 11 |- ( h = ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) -> ( ( h ` n ) = 1o <-> ( ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) ` n ) = 1o ) )
3433rexbidv 2471 . . . . . . . . . 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 662 . . . . . . . . 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 662 . . . . . . . 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 233 . . . . . . 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 7131 . . . . . . . . . 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 146 . . . . . . . 8 |- ( ph -> A. h e. ( 2o ^m om ) ( -. -. E. n e. om ( h ` n ) = 1o -> E. n e. om ( h ` n ) = 1o ) )
4241adantr 274 . . . . . . 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 6421 . . . . . . . . . . . 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 6420 . . . . . . . . . . . 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 525 . . . . . . . . . . . . . . 15 |- ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ n e. om ) -> g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) )
48 fof 5420 . . . . . . . . . . . . . . 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 109 . . . . . . . . . . . . . 14 |- ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ n e. om ) -> n e. om )
5149, 50ffvelrnd 5632 . . . . . . . . . . . . 13 |- ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ n e. om ) -> ( g ` n ) e. ( { z e. om | y = { (/) } } ⊔ 1o ) )
52 eldju1st 7048 . . . . . . . . . . . . 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 6411 . . . . . . . . . . . . . . . 16 |- 1o =/= (/)
5554neii 2342 . . . . . . . . . . . . . . 15 |- -. 1o = (/)
56 eqeq1 2177 . . . . . . . . . . . . . . 15 |- ( ( 1st ` ( g ` n ) ) = 1o -> ( ( 1st ` ( g ` n ) ) = (/) <-> 1o = (/) ) )
5755, 56mtbiri 670 . . . . . . . . . . . . . 14 |- ( ( 1st ` ( g ` n ) ) = 1o -> -. ( 1st ` ( g ` n ) ) = (/) )
5857orim2i 756 . . . . . . . . . . . . 13 |- ( ( ( 1st ` ( g ` n ) ) = (/) \/ ( 1st ` ( g ` n ) ) = 1o ) -> ( ( 1st ` ( g ` n ) ) = (/) \/ -. ( 1st ` ( g ` n ) ) = (/) ) )
59 df-dc 830 . . . . . . . . . . . . 13 |- (DECID ( 1st ` ( g ` n ) ) = (/) <-> ( ( 1st ` ( g ` n ) ) = (/) \/ -. ( 1st ` ( g ` n ) ) = (/) ) )
6058, 59sylibr 133 . . . . . . . . . . . 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 3555 . . . . . . . . . 10 |- ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ n e. om ) -> if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) e. 2o )
6362fmpttd 5651 . . . . . . . . 9 |- ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) -> ( n e. om |-> if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) ) : om --> 2o )
64 2fveq3 5501 . . . . . . . . . . . . . 14 |- ( w = n -> ( 1st ` ( g ` w ) ) = ( 1st ` ( g ` n ) ) )
6564eqeq1d 2179 . . . . . . . . . . . . 13 |- ( w = n -> ( ( 1st ` ( g ` w ) ) = (/) <-> ( 1st ` ( g ` n ) ) = (/) ) )
6665ifbid 3547 . . . . . . . . . . . 12 |- ( w = n -> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) = if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) )
67 eqcom 2172 . . . . . . . . . . . 12 |- ( w = n <-> n = w )
68 eqcom 2172 . . . . . . . . . . . 12 |- ( if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) = if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) <-> if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) = if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) )
6966, 67, 683imtr3i 199 . . . . . . . . . . 11 |- ( n = w -> if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) = if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) )
7069cbvmptv 4085 . . . . . . . . . 10 |- ( n e. om |-> if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) ) = ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) )
7170feq1i 5340 . . . . . . . . 9 |- ( ( n e. om |-> if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) ) : om --> 2o <-> ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) : om --> 2o )
7263, 71sylib 121 . . . . . . . 8 |- ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) -> ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) : om --> 2o )
73 2onn 6500 . . . . . . . . . 10 |- 2o e. om
7473elexi 2742 . . . . . . . . 9 |- 2o e. _V
7574, 2elmap 6655 . . . . . . . 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 133 . . . . . . 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 2839 . . . . . 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 2170 . . . . . . . . . . . . 13 |- ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) = ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) )
7978, 66, 50, 62fvmptd3 5589 . . . . . . . . . . . 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 2179 . . . . . . . . . . 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 274 . . . . . . . . . . . . . . 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 109 . . . . . . . . . . . . . . . . 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 2176 . . . . . . . . . . . . . . . 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 3560 . . . . . . . . . . . . . . . . . . 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 2170 . . . . . . . . . . . . . . . . . . 19 |- 1o = 1o
87 orcom 723 . . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( 1st ` ( g ` n ) ) = (/) /\ 1o = 1o ) \/ ( -. ( 1st ` ( g ` n ) ) = (/) /\ 1o = (/) ) ) <-> ( ( -. ( 1st ` ( g ` n ) ) = (/) /\ 1o = (/) ) \/ ( ( 1st ` ( g ` n ) ) = (/) /\ 1o = 1o ) ) )
8855intnan 924 . . . . . . . . . . . . . . . . . . . . 21 |- -. ( -. ( 1st ` ( g ` n ) ) = (/) /\ 1o = (/) )
89 biorf 739 . . . . . . . . . . . . . . . . . . . . 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 186 . . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( 1st ` ( g ` n ) ) = (/) /\ 1o = 1o ) \/ ( -. ( 1st ` ( g ` n ) ) = (/) /\ 1o = (/) ) ) <-> ( ( 1st ` ( g ` n ) ) = (/) /\ 1o = 1o ) )
9286, 91mpbiran2 936 . . . . . . . . . . . . . . . . . 18 |- ( ( ( ( 1st ` ( g ` n ) ) = (/) /\ 1o = 1o ) \/ ( -. ( 1st ` ( g ` n ) ) = (/) /\ 1o = (/) ) ) <-> ( 1st ` ( g ` n ) ) = (/) )
9385, 92bitrdi 195 . . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ n e. om ) -> ( 1o = if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) <-> ( 1st ` ( g ` n ) ) = (/) ) )
9493adantr 274 . . . . . . . . . . . . . . . 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 146 . . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ n e. om ) /\ if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) = 1o ) -> ( 1st ` ( g ` n ) ) = (/) )
96 eldju2ndl 7049 . . . . . . . . . . . . . . 15 |- ( ( ( g ` n ) e. ( { z e. om | y = { (/) } } ⊔ 1o ) /\ ( 1st ` ( g ` n ) ) = (/) ) -> ( 2nd ` ( g ` n ) ) e. { z e. om | y = { (/) } } )
9781, 95, 96syl2anc 409 . . . . . . . . . . . . . 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 171 . . . . . . . . . . . . . . 15 |- ( z = ( 2nd ` ( g ` n ) ) -> ( y = { (/) } <-> y = { (/) } ) )
9998elrab 2886 . . . . . . . . . . . . . 14 |- ( ( 2nd ` ( g ` n ) ) e. { z e. om | y = { (/) } } <-> ( ( 2nd ` ( g ` n ) ) e. om /\ y = { (/) } ) )
10097, 99sylib 121 . . . . . . . . . . . . 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 113 . . . . . . . . . . . 12 |- ( ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ n e. om ) /\ if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) = 1o ) -> y = { (/) } )
102101ex 114 . . . . . . . . . . 11 |- ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ n e. om ) -> ( if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) = 1o -> y = { (/) } ) )
10380, 102sylbid 149 . . . . . . . . . 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 2587 . . . . . . . . 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 525 . . . . . . . . . . . 12 |- ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ y = { (/) } ) -> g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) )
106 biidd 171 . . . . . . . . . . . . . 14 |- ( z = (/) -> ( y = { (/) } <-> y = { (/) } ) )
107 peano1 4578 . . . . . . . . . . . . . . 15 |- (/) e. om
108107a1i 9 . . . . . . . . . . . . . 14 |- ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ y = { (/) } ) -> (/) e. om )
109 simpr 109 . . . . . . . . . . . . . 14 |- ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ y = { (/) } ) -> y = { (/) } )
110106, 108, 109elrabd 2888 . . . . . . . . . . . . 13 |- ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ y = { (/) } ) -> (/) e. { z e. om | y = { (/) } } )
111 djulcl 7028 . . . . . . . . . . . . 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 5732 . . . . . . . . . . . 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 409 . . . . . . . . . . 11 |- ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ y = { (/) } ) -> E. n e. om (inl ` (/) ) = ( g ` n ) )
11579adantlr 474 . . . . . . . . . . . . . 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 5496 . . . . . . . . . . . . . . . 16 |- ( (inl ` (/) ) = ( g ` n ) -> ( 1st ` (inl ` (/) ) ) = ( 1st ` ( g ` n ) ) )
117 1stinl 7051 . . . . . . . . . . . . . . . . 17 |- ( (/) e. om -> ( 1st ` (inl ` (/) ) ) = (/) )
118107, 117ax-mp 5 . . . . . . . . . . . . . . . 16 |- ( 1st ` (inl ` (/) ) ) = (/)
119116, 118eqtr3di 2218 . . . . . . . . . . . . . . 15 |- ( (inl ` (/) ) = ( g ` n ) -> ( 1st ` ( g ` n ) ) = (/) )
120119iftrued 3533 . . . . . . . . . . . . . 14 |- ( (inl ` (/) ) = ( g ` n ) -> if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) = 1o )
121115, 120sylan9eq 2223 . . . . . . . . . . . . 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 114 . . . . . . . . . . . 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 2572 . . . . . . . . . . 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 114 . . . . . . . . 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 128 . . . . . . . 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 662 . . . . . . 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 662 . . . . . 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 201 . . . . 5 |- ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) -> ( -. -. y = { (/) } -> y = { (/) } ) )
130 df-stab 826 . . . . 5 |- (STAB y = { (/) } <-> ( -. -. y = { (/) } -> y = { (/) } ) )
131129, 130sylibr 133 . . . 4 |- ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) -> STAB y = { (/) } )
13231, 131exlimddv 1891 . . 3 |- ( ph -> STAB y = { (/) } )
133132adantr 274 . 2 |- ( ( ph /\ y C_ { (/) } ) -> STAB y = { (/) } )
134133exmid1stab 14033 1 |- ( ph -> EXMID )
Colors of variables: wff set class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 703  STAB wstab 825  DECID wdc 829   A.wal 1346   = wceq 1348   E.wex 1485   e. wcel 2141   A.wral 2448   E.wrex 2449   {crab 2452   _Vcvv 2730   C_ wss 3121   (/)c0 3414   ifcif 3526   {csn 3583   |-> cmpt 4050  EXMIDwem 4180   _I cid 4273   omcom 4574   |` cres 4613   -->wf 5194   -onto->wfo 5196   -1-1-onto->wf1o 5197   ` cfv 5198  (class class class)co 5853   1stc1st 6117   2ndc2nd 6118   1oc1o 6388   2oc2o 6389   ^m cmap 6626   ⊔ cdju 7014  inlcinl 7022  Markovcmarkov 7127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-exmid 4181  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-1o 6395  df-2o 6396  df-map 6628  df-dju 7015  df-inl 7024  df-inr 7025  df-markov 7128
This theorem is referenced by: (None)
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