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Theorem subctctexmid 17013
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 4738 . . . . . . . 8 |- om e. _V
32rabex 4278 . . . . . . 7 |- { z e. om | y = { (/) } } e. _V
43a1i 9 . . . . . 6 |- ( ph -> { z e. om | y = { (/) } } e. _V )
5 ssrab2 3333 . . . . . . 7 |- { z e. om | y = { (/) } } C_ om
6 f1oi 5677 . . . . . . . . 9 |- ( _I |` { z e. om | y = { (/) } } ) : { z e. om | y = { (/) } } -1-1-onto-> { z e. om | y = { (/) } }
7 f1ofo 5644 . . . . . . . . 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 5106 . . . . . . . . . 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 5609 . . . . . . . . 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 2920 . . . . . . . 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 3271 . . . . . . . 8 |- ( s = { z e. om | y = { (/) } } -> ( s C_ om <-> { z e. om | y = { (/) } } C_ om ) )
16 foeq2 5610 . . . . . . . . 9 |- ( s = { z e. om | y = { (/) } } -> ( f : s -onto-> { z e. om | y = { (/) } } <-> f : { z e. om | y = { (/) } } -onto-> { z e. om | y = { (/) } } ) )
1716exbidv 1878 . . . . . . . 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 477 . . . . . . 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 2913 . . . . . 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 1496 . . . . 5 |- ( ph -> E. s ( s C_ om /\ E. f f : s -onto-> { z e. om | y = { (/) } } ) )
21 foeq3 5611 . . . . . . . . . 10 |- ( x = { z e. om | y = { (/) } } -> ( f : s -onto-> x <-> f : s -onto-> { z e. om | y = { (/) } } ) )
2221exbidv 1878 . . . . . . . . 9 |- ( x = { z e. om | y = { (/) } } -> ( E. f f : s -onto-> x <-> E. f f : s -onto-> { z e. om | y = { (/) } } ) )
2322anbi2d 468 . . . . . . . 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 1878 . . . . . . 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 7374 . . . . . . . . 9 |- ( x = { z e. om | y = { (/) } } -> ( x ⊔ 1o ) = ( { z e. om | y = { (/) } } ⊔ 1o ) )
26 foeq3 5611 . . . . . . . . 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 1878 . . . . . . 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 2919 . . . . 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 5692 . . . . . . . . . . . 12 |- ( h = ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) -> ( h ` n ) = ( ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) ` n ) )
3332eqeq1d 2247 . . . . . . . . . . 11 |- ( h = ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) -> ( ( h ` n ) = 1o <-> ( ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) ` n ) = 1o ) )
3433rexbidv 2551 . . . . . . . . . 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 677 . . . . . . . . 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 677 . . . . . . . 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 7489 . . . . . . . . . 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 6709 . . . . . . . . . . . 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 6708 . . . . . . . . . . . 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 533 . . . . . . . . . . . . . . 15 |- ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ n e. om ) -> g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) )
48 fof 5613 . . . . . . . . . . . . . . 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 5838 . . . . . . . . . . . . 13 |- ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ n e. om ) -> ( g ` n ) e. ( { z e. om | y = { (/) } } ⊔ 1o ) )
52 eldju1st 7405 . . . . . . . . . . . . 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 6699 . . . . . . . . . . . . . . . 16 |- 1o =/= (/)
5554neii 2422 . . . . . . . . . . . . . . 15 |- -. 1o = (/)
56 eqeq1 2245 . . . . . . . . . . . . . . 15 |- ( ( 1st ` ( g ` n ) ) = 1o -> ( ( 1st ` ( g ` n ) ) = (/) <-> 1o = (/) ) )
5755, 56mtbiri 686 . . . . . . . . . . . . . 14 |- ( ( 1st ` ( g ` n ) ) = 1o -> -. ( 1st ` ( g ` n ) ) = (/) )
5857orim2i 773 . . . . . . . . . . . . 13 |- ( ( ( 1st ` ( g ` n ) ) = (/) \/ ( 1st ` ( g ` n ) ) = 1o ) -> ( ( 1st ` ( g ` n ) ) = (/) \/ -. ( 1st ` ( g ` n ) ) = (/) ) )
59 df-dc 847 . . . . . . . . . . . . 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 3670 . . . . . . . . . 10 |- ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ n e. om ) -> if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) e. 2o )
6362fmpttd 5857 . . . . . . . . 9 |- ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) -> ( n e. om |-> if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) ) : om --> 2o )
64 2fveq3 5698 . . . . . . . . . . . . . 14 |- ( w = n -> ( 1st ` ( g ` w ) ) = ( 1st ` ( g ` n ) ) )
6564eqeq1d 2247 . . . . . . . . . . . . 13 |- ( w = n -> ( ( 1st ` ( g ` w ) ) = (/) <-> ( 1st ` ( g ` n ) ) = (/) ) )
6665ifbid 3662 . . . . . . . . . . . 12 |- ( w = n -> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) = if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) )
67 eqcom 2240 . . . . . . . . . . . 12 |- ( w = n <-> n = w )
68 eqcom 2240 . . . . . . . . . . . 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 4225 . . . . . . . . . 10 |- ( n e. om |-> if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) ) = ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) )
7170feq1i 5524 . . . . . . . . 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 6788 . . . . . . . . . 10 |- 2o e. om
7473elexi 2834 . . . . . . . . 9 |- 2o e. _V
7574, 2elmap 6952 . . . . . . . 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 2934 . . . . . 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 2238 . . . . . . . . . . . . 13 |- ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) ) = ( w e. om |-> if ( ( 1st ` ( g ` w ) ) = (/) , 1o , (/) ) )
7978, 66, 50, 62fvmptd3 5796 . . . . . . . . . . . 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 2247 . . . . . . . . . . 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 2244 . . . . . . . . . . . . . . . 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 3677 . . . . . . . . . . . . . . . . . . 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 2238 . . . . . . . . . . . . . . . . . . 19 |- 1o = 1o
87 orcom 740 . . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( 1st ` ( g ` n ) ) = (/) /\ 1o = 1o ) \/ ( -. ( 1st ` ( g ` n ) ) = (/) /\ 1o = (/) ) ) <-> ( ( -. ( 1st ` ( g ` n ) ) = (/) /\ 1o = (/) ) \/ ( ( 1st ` ( g ` n ) ) = (/) /\ 1o = 1o ) ) )
8855intnan 941 . . . . . . . . . . . . . . . . . . . . 21 |- -. ( -. ( 1st ` ( g ` n ) ) = (/) /\ 1o = (/) )
89 biorf 756 . . . . . . . . . . . . . . . . . . . . 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 954 . . . . . . . . . . . . . . . . . 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 7406 . . . . . . . . . . . . . . 15 |- ( ( ( g ` n ) e. ( { z e. om | y = { (/) } } ⊔ 1o ) /\ ( 1st ` ( g ` n ) ) = (/) ) -> ( 2nd ` ( g ` n ) ) e. { z e. om | y = { (/) } } )
9781, 95, 96syl2anc 415 . . . . . . . . . . . . . 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 2982 . . . . . . . . . . . . . 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 2668 . . . . . . . . 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 533 . . . . . . . . . . . 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 4739 . . . . . . . . . . . . . . 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 2984 . . . . . . . . . . . . 13 |- ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ y = { (/) } ) -> (/) e. { z e. om | y = { (/) } } )
111 djulcl 7385 . . . . . . . . . . . . 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 5952 . . . . . . . . . . . 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 415 . . . . . . . . . . 11 |- ( ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) /\ y = { (/) } ) -> E. n e. om (inl ` (/) ) = ( g ` n ) )
11579adantlr 481 . . . . . . . . . . . . . 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 5693 . . . . . . . . . . . . . . . 16 |- ( (inl ` (/) ) = ( g ` n ) -> ( 1st ` (inl ` (/) ) ) = ( 1st ` ( g ` n ) ) )
117 1stinl 7408 . . . . . . . . . . . . . . . . 17 |- ( (/) e. om -> ( 1st ` (inl ` (/) ) ) = (/) )
118107, 117ax-mp 5 . . . . . . . . . . . . . . . 16 |- ( 1st ` (inl ` (/) ) ) = (/)
119116, 118eqtr3di 2286 . . . . . . . . . . . . . . 15 |- ( (inl ` (/) ) = ( g ` n ) -> ( 1st ` ( g ` n ) ) = (/) )
120119iftrued 3647 . . . . . . . . . . . . . 14 |- ( (inl ` (/) ) = ( g ` n ) -> if ( ( 1st ` ( g ` n ) ) = (/) , 1o , (/) ) = 1o )
121115, 120sylan9eq 2291 . . . . . . . . . . . . 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 2652 . . . . . . . . . . 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 677 . . . . . . 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 677 . . . . . 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 843 . . . . 5 |- (STAB y = { (/) } <-> ( -. -. y = { (/) } -> y = { (/) } ) )
131129, 130sylibr 134 . . . 4 |- ( ( ph /\ g : om -onto-> ( { z e. om | y = { (/) } } ⊔ 1o ) ) -> STAB y = { (/) } )
13231, 131exlimddv 1954 . . 3 |- ( ph -> STAB y = { (/) } )
133132adantr 276 . 2 |- ( ( ph /\ y C_ { (/) } ) -> STAB y = { (/) } )
134133exmid1stab 4343 1 |- ( ph -> EXMID )
Colors of variables: wff set class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720  STAB wstab 842  DECID wdc 846   A.wal 1400   = wceq 1402   E.wex 1545   e. wcel 2209   A.wral 2528   E.wrex 2529   {crab 2532   _Vcvv 2821   C_ wss 3220   (/)c0 3520   ifcif 3638   {csn 3708   |-> cmpt 4190  EXMIDwem 4329   _I cid 4431   omcom 4735   |` cres 4774   -->wf 5371   -onto->wfo 5373   -1-1-onto->wf1o 5374   ` cfv 5375  (class class class)co 6079   1stc1st 6366   2ndc2nd 6367   1oc1o 6674   2oc2o 6675   ^m cmap 6916   ⊔ cdju 7371  inlcinl 7379  Markovcmarkov 7485
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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733
This theorem depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-exmid 4330  df-id 4436  df-iord 4509  df-on 4511  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-1o 6681  df-2o 6682  df-map 6918  df-dju 7372  df-inl 7381  df-inr 7382  df-markov 7486
This theorem is referenced by: (None)
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