Theorem onintexmid 4482

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 3673	. . . . . 6 |- ( ( u e. On /\ v e. On ) -> { u , v } C_ On )
2		prmg 3639	. . . . . . 7 |- ( u e. On -> E. x x e. { u , v } )
3	2	adantr 274	. . . . . 6 |- ( ( u e. On /\ v e. On ) -> E. x x e. { u , v } )
4		zfpair2 4127	. . . . . . 7 |- { u , v } e. _V
5		sseq1 3115	. . . . . . . . 9 |- ( y = { u , v } -> ( y C_ On <-> { u , v } C_ On ) )
6		eleq2 2201	. . . . . . . . . 10 |- ( y = { u , v } -> ( x e. y <-> x e. { u , v } ) )
7	6	exbidv 1797	. . . . . . . . 9 |- ( y = { u , v } -> ( E. x x e. y <-> E. x x e. { u , v } ) )
8	5, 7	anbi12d 464	. . . . . . . 8 |- ( y = { u , v } -> ( ( y C_ On /\ E. x x e. y ) <-> ( { u , v } C_ On /\ E. x x e. { u , v } ) ) )
9		inteq 3769	. . . . . . . . 9 |- ( y = { u , v } -> |^| y = |^| { u , v } )
10		id 19	. . . . . . . . 9 |- ( y = { u , v } -> y = { u , v } )
11	9, 10	eleq12d 2208	. . . . . . . 8 |- ( y = { u , v } -> ( |^| y e. y <-> |^| { u , v } e. { u , v } ) )
12	8, 11	imbi12d 233	. . . . . . 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 2735	. . . . . 6 |- ( ( { u , v } C_ On /\ E. x x e. { u , v } ) -> |^| { u , v } e. { u , v } )
15	1, 3, 14	syl2anc 408	. . . . 5 |- ( ( u e. On /\ v e. On ) -> |^| { u , v } e. { u , v } )
16		elpri 3545	. . . . 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 3263	. . . . . . 7 |- ( v i^i u ) = ( u i^i v )
19	18	eqeq1i 2145	. . . . . 6 |- ( ( v i^i u ) = u <-> ( u i^i v ) = u )
20		dfss1 3275	. . . . . 6 |- ( u C_ v <-> ( v i^i u ) = u )
21		vex 2684	. . . . . . . 8 |- u e. _V
22		vex 2684	. . . . . . . 8 |- v e. _V
23	21, 22	intpr 3798	. . . . . . 7 |- |^| { u , v } = ( u i^i v )
24	23	eqeq1i 2145	. . . . . 6 |- ( |^| { u , v } = u <-> ( u i^i v ) = u )
25	19, 20, 24	3bitr4ri 212	. . . . 5 |- ( |^| { u , v } = u <-> u C_ v )
26	23	eqeq1i 2145	. . . . . 6 |- ( |^| { u , v } = v <-> ( u i^i v ) = v )
27		dfss1 3275	. . . . . 6 |- ( v C_ u <-> ( u i^i v ) = v )
28	26, 27	bitr4i 186	. . . . 5 |- ( |^| { u , v } = v <-> v C_ u )
29	25, 28	orbi12i 753	. . . 4 |- ( ( |^| { u , v } = u \/ |^| { u , v } = v ) <-> ( u C_ v \/ v C_ u ) )
30	17, 29	sylib 121	. . 3 |- ( ( u e. On /\ v e. On ) -> ( u C_ v \/ v C_ u ) )
31	30	rgen2a 2484	. 2 |- A. u e. On A. v e. On ( u C_ v \/ v C_ u )
32	31	ordtri2or2exmid 4481	1 |- ( ph \/ -. ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 697   = wceq 1331   E.wex 1468   e. wcel 1480   i^i cin 3065   C_ wss 3066   {cpr 3523   |^|cint 3766   Oncon0 4280

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-uni 3732  df-int 3767  df-tr 4022  df-iord 4283  df-on 4285  df-suc 4288

This theorem is referenced by: (None)