Theorem onintexmid 4572

Description: If the intersection (infimum) of an inhabited class of ordinal numbers belongs to the class, excluded middle follows. The hypothesis would be provable given excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Aug-2021.)

Hypothesis

Ref	Expression
onintexmid.onint	|- ( ( y C_ On /\ E. x x e. y ) -> |^| y e. y )

Assertion

Ref	Expression
onintexmid	|- ( ph \/ -. ph )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem onintexmid

Dummy variables u v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prssi 3750	. . . . . 6 |- ( ( u e. On /\ v e. On ) -> { u , v } C_ On )
2		prmg 3713	. . . . . . 7 |- ( u e. On -> E. x x e. { u , v } )
3	2	adantr 276	. . . . . 6 |- ( ( u e. On /\ v e. On ) -> E. x x e. { u , v } )
4		zfpair2 4210	. . . . . . 7 |- { u , v } e. _V
5		sseq1 3178	. . . . . . . . 9 |- ( y = { u , v } -> ( y C_ On <-> { u , v } C_ On ) )
6		eleq2 2241	. . . . . . . . . 10 |- ( y = { u , v } -> ( x e. y <-> x e. { u , v } ) )
7	6	exbidv 1825	. . . . . . . . 9 |- ( y = { u , v } -> ( E. x x e. y <-> E. x x e. { u , v } ) )
8	5, 7	anbi12d 473	. . . . . . . 8 |- ( y = { u , v } -> ( ( y C_ On /\ E. x x e. y ) <-> ( { u , v } C_ On /\ E. x x e. { u , v } ) ) )
9		inteq 3847	. . . . . . . . 9 |- ( y = { u , v } -> |^| y = |^| { u , v } )
10		id 19	. . . . . . . . 9 |- ( y = { u , v } -> y = { u , v } )
11	9, 10	eleq12d 2248	. . . . . . . 8 |- ( y = { u , v } -> ( |^| y e. y <-> |^| { u , v } e. { u , v } ) )
12	8, 11	imbi12d 234	. . . . . . 7 |- ( y = { u , v } -> ( ( ( y C_ On /\ E. x x e. y ) -> |^| y e. y ) <-> ( ( { u , v } C_ On /\ E. x x e. { u , v } ) -> |^| { u , v } e. { u , v } ) ) )
13		onintexmid.onint	. . . . . . 7 |- ( ( y C_ On /\ E. x x e. y ) -> |^| y e. y )
14	4, 12, 13	vtocl 2791	. . . . . 6 |- ( ( { u , v } C_ On /\ E. x x e. { u , v } ) -> |^| { u , v } e. { u , v } )
15	1, 3, 14	syl2anc 411	. . . . 5 |- ( ( u e. On /\ v e. On ) -> |^| { u , v } e. { u , v } )
16		elpri 3615	. . . . 5 |- ( |^| { u , v } e. { u , v } -> ( |^| { u , v } = u \/ |^| { u , v } = v ) )
17	15, 16	syl 14	. . . 4 |- ( ( u e. On /\ v e. On ) -> ( |^| { u , v } = u \/ |^| { u , v } = v ) )
18		incom 3327	. . . . . . 7 |- ( v i^i u ) = ( u i^i v )
19	18	eqeq1i 2185	. . . . . 6 |- ( ( v i^i u ) = u <-> ( u i^i v ) = u )
20		dfss1 3339	. . . . . 6 |- ( u C_ v <-> ( v i^i u ) = u )
21		vex 2740	. . . . . . . 8 |- u e. _V
22		vex 2740	. . . . . . . 8 |- v e. _V
23	21, 22	intpr 3876	. . . . . . 7 |- |^| { u , v } = ( u i^i v )
24	23	eqeq1i 2185	. . . . . 6 |- ( |^| { u , v } = u <-> ( u i^i v ) = u )
25	19, 20, 24	3bitr4ri 213	. . . . 5 |- ( |^| { u , v } = u <-> u C_ v )
26	23	eqeq1i 2185	. . . . . 6 |- ( |^| { u , v } = v <-> ( u i^i v ) = v )
27		dfss1 3339	. . . . . 6 |- ( v C_ u <-> ( u i^i v ) = v )
28	26, 27	bitr4i 187	. . . . 5 |- ( |^| { u , v } = v <-> v C_ u )
29	25, 28	orbi12i 764	. . . 4 |- ( ( |^| { u , v } = u \/ |^| { u , v } = v ) <-> ( u C_ v \/ v C_ u ) )
30	17, 29	sylib 122	. . 3 |- ( ( u e. On /\ v e. On ) -> ( u C_ v \/ v C_ u ) )
31	30	rgen2a 2531	. 2 |- A. u e. On A. v e. On ( u C_ v \/ v C_ u )
32	31	ordtri2or2exmid 4570	1 |- ( ph \/ -. ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 708   = wceq 1353   E.wex 1492   e. wcel 2148   i^i cin 3128   C_ wss 3129   {cpr 3593   |^|cint 3844   Oncon0 4363

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-uni 3810  df-int 3845  df-tr 4102  df-iord 4366  df-on 4368  df-suc 4371

This theorem is referenced by: (None)