Theorem onintexmid 4626

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 3794	. . . . . 6 |- ( ( u e. On /\ v e. On ) -> { u , v } C_ On )
2		prmg 3757	. . . . . . 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 4259	. . . . . . 7 |- { u , v } e. _V
5		sseq1 3218	. . . . . . . . 9 |- ( y = { u , v } -> ( y C_ On <-> { u , v } C_ On ) )
6		eleq2 2270	. . . . . . . . . 10 |- ( y = { u , v } -> ( x e. y <-> x e. { u , v } ) )
7	6	exbidv 1849	. . . . . . . . 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 3891	. . . . . . . . 9 |- ( y = { u , v } -> |^| y = |^| { u , v } )
10		id 19	. . . . . . . . 9 |- ( y = { u , v } -> y = { u , v } )
11	9, 10	eleq12d 2277	. . . . . . . 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 2829	. . . . . 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 3658	. . . . 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 3367	. . . . . . 7 |- ( v i^i u ) = ( u i^i v )
19	18	eqeq1i 2214	. . . . . 6 |- ( ( v i^i u ) = u <-> ( u i^i v ) = u )
20		dfss1 3379	. . . . . 6 |- ( u C_ v <-> ( v i^i u ) = u )
21		vex 2776	. . . . . . . 8 |- u e. _V
22		vex 2776	. . . . . . . 8 |- v e. _V
23	21, 22	intpr 3920	. . . . . . 7 |- |^| { u , v } = ( u i^i v )
24	23	eqeq1i 2214	. . . . . 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 2214	. . . . . 6 |- ( |^| { u , v } = v <-> ( u i^i v ) = v )
27		dfss1 3379	. . . . . 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 766	. . . 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 2561	. 2 |- A. u e. On A. v e. On ( u C_ v \/ v C_ u )
32	31	ordtri2or2exmid 4624	1 |- ( ph \/ -. ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 710   = wceq 1373   E.wex 1516   e. wcel 2177   i^i cin 3167   C_ wss 3168   {cpr 3636   |^|cint 3888   Oncon0 4415

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4167  ax-nul 4175  ax-pow 4223  ax-pr 4258  ax-un 4485  ax-setind 4590

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-dif 3170  df-un 3172  df-in 3174  df-ss 3181  df-nul 3463  df-pw 3620  df-sn 3641  df-pr 3642  df-uni 3854  df-int 3889  df-tr 4148  df-iord 4418  df-on 4420  df-suc 4423

This theorem is referenced by: (None)