Theorem onintexmid 4495

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 3686	. . . . . 6 |- ( ( u e. On /\ v e. On ) -> { u , v } C_ On )
2		prmg 3652	. . . . . . 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 4140	. . . . . . 7 |- { u , v } e. _V
5		sseq1 3125	. . . . . . . . 9 |- ( y = { u , v } -> ( y C_ On <-> { u , v } C_ On ) )
6		eleq2 2204	. . . . . . . . . 10 |- ( y = { u , v } -> ( x e. y <-> x e. { u , v } ) )
7	6	exbidv 1798	. . . . . . . . 9 |- ( y = { u , v } -> ( E. x x e. y <-> E. x x e. { u , v } ) )
8	5, 7	anbi12d 465	. . . . . . . 8 |- ( y = { u , v } -> ( ( y C_ On /\ E. x x e. y ) <-> ( { u , v } C_ On /\ E. x x e. { u , v } ) ) )
9		inteq 3782	. . . . . . . . 9 |- ( y = { u , v } -> |^| y = |^| { u , v } )
10		id 19	. . . . . . . . 9 |- ( y = { u , v } -> y = { u , v } )
11	9, 10	eleq12d 2211	. . . . . . . 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 2743	. . . . . 6 |- ( ( { u , v } C_ On /\ E. x x e. { u , v } ) -> |^| { u , v } e. { u , v } )
15	1, 3, 14	syl2anc 409	. . . . 5 |- ( ( u e. On /\ v e. On ) -> |^| { u , v } e. { u , v } )
16		elpri 3555	. . . . 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 3273	. . . . . . 7 |- ( v i^i u ) = ( u i^i v )
19	18	eqeq1i 2148	. . . . . 6 |- ( ( v i^i u ) = u <-> ( u i^i v ) = u )
20		dfss1 3285	. . . . . 6 |- ( u C_ v <-> ( v i^i u ) = u )
21		vex 2692	. . . . . . . 8 |- u e. _V
22		vex 2692	. . . . . . . 8 |- v e. _V
23	21, 22	intpr 3811	. . . . . . 7 |- |^| { u , v } = ( u i^i v )
24	23	eqeq1i 2148	. . . . . 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 2148	. . . . . 6 |- ( |^| { u , v } = v <-> ( u i^i v ) = v )
27		dfss1 3285	. . . . . 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 754	. . . 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 2489	. 2 |- A. u e. On A. v e. On ( u C_ v \/ v C_ u )
32	31	ordtri2or2exmid 4494	1 |- ( ph \/ -. ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 698   = wceq 1332   E.wex 1469   e. wcel 1481   i^i cin 3075   C_ wss 3076   {cpr 3533   |^|cint 3779   Oncon0 4293

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-uni 3745  df-int 3780  df-tr 4035  df-iord 4296  df-on 4298  df-suc 4301

This theorem is referenced by: (None)