Theorem onintexmid 4487

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 3678	. . . . . 6 |- ( ( u e. On /\ v e. On ) -> { u , v } C_ On )
2		prmg 3644	. . . . . . 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 4132	. . . . . . 7 |- { u , v } e. _V
5		sseq1 3120	. . . . . . . . 9 |- ( y = { u , v } -> ( y C_ On <-> { u , v } C_ On ) )
6		eleq2 2203	. . . . . . . . . 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 3774	. . . . . . . . 9 |- ( y = { u , v } -> |^| y = |^| { u , v } )
10		id 19	. . . . . . . . 9 |- ( y = { u , v } -> y = { u , v } )
11	9, 10	eleq12d 2210	. . . . . . . 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 2740	. . . . . 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 3550	. . . . 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 3268	. . . . . . 7 |- ( v i^i u ) = ( u i^i v )
19	18	eqeq1i 2147	. . . . . 6 |- ( ( v i^i u ) = u <-> ( u i^i v ) = u )
20		dfss1 3280	. . . . . 6 |- ( u C_ v <-> ( v i^i u ) = u )
21		vex 2689	. . . . . . . 8 |- u e. _V
22		vex 2689	. . . . . . . 8 |- v e. _V
23	21, 22	intpr 3803	. . . . . . 7 |- |^| { u , v } = ( u i^i v )
24	23	eqeq1i 2147	. . . . . 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 2147	. . . . . 6 |- ( |^| { u , v } = v <-> ( u i^i v ) = v )
27		dfss1 3280	. . . . . 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 2486	. 2 |- A. u e. On A. v e. On ( u C_ v \/ v C_ u )
32	31	ordtri2or2exmid 4486	1 |- ( ph \/ -. ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 697   = wceq 1331   E.wex 1468   e. wcel 1480   i^i cin 3070   C_ wss 3071   {cpr 3528   |^|cint 3771   Oncon0 4285

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 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-uni 3737  df-int 3772  df-tr 4027  df-iord 4288  df-on 4290  df-suc 4293

This theorem is referenced by: (None)