Theorem acexmidlema 5503

Description: Lemma for acexmid 5511. (Contributed by Jim Kingdon, 6-Aug-2019.)

Hypotheses

Ref	Expression
acexmidlem.a	|- A = { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) }
acexmidlem.b	|- B = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ph ) }
acexmidlem.c	|- C = { A , B }

Assertion

Ref	Expression
acexmidlema	|- ( { (/) } e. A -> ph )

Distinct variable groups:   x, A   x, B   x, C   ph, x

Proof of Theorem acexmidlema

Step	Hyp	Ref	Expression
1		acexmidlem.a	. . . 4 |- A = { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) }
2	1	eleq2i 2104	. . 3 |- ( { (/) } e. A <-> { (/) } e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } )
3		p0ex 3939	. . . . 5 |- { (/) } e. _V
4	3	prid2 3477	. . . 4 |- { (/) } e. { (/) , { (/) } }
5		eqeq1 2046	. . . . . 6 |- ( x = { (/) } -> ( x = (/) <-> { (/) } = (/) ) )
6	5	orbi1d 705	. . . . 5 |- ( x = { (/) } -> ( ( x = (/) \/ ph ) <-> ( { (/) } = (/) \/ ph ) ) )
7	6	elrab3 2699	. . . 4 |- ( { (/) } e. { (/) , { (/) } } -> ( { (/) } e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } <-> ( { (/) } = (/) \/ ph ) ) )
8	4, 7	ax-mp 7	. . 3 |- ( { (/) } e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } <-> ( { (/) } = (/) \/ ph ) )
9	2, 8	bitri 173	. 2 |- ( { (/) } e. A <-> ( { (/) } = (/) \/ ph ) )
10		noel 3228	. . . 4 |- -. (/) e. (/)
11		0ex 3884	. . . . . 6 |- (/) e. _V
12	11	snid 3402	. . . . 5 |- (/) e. { (/) }
13		eleq2 2101	. . . . 5 |- ( { (/) } = (/) -> ( (/) e. { (/) } <-> (/) e. (/) ) )
14	12, 13	mpbii 136	. . . 4 |- ( { (/) } = (/) -> (/) e. (/) )
15	10, 14	mto 588	. . 3 |- -. { (/) } = (/)
16		orel1 644	. . 3 |- ( -. { (/) } = (/) -> ( ( { (/) } = (/) \/ ph ) -> ph ) )
17	15, 16	ax-mp 7	. 2 |- ( ( { (/) } = (/) \/ ph ) -> ph )
18	9, 17	sylbi 114	1 |- ( { (/) } e. A -> ph )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 98   \/ wo 629   = wceq 1243   e. wcel 1393   {crab 2310   (/)c0 3224   {csn 3375   {cpr 3376

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382

This theorem is referenced by:  acexmidlem1  5508