Theorem acexmidlemb 5880

Description: Lemma for acexmid 5887. (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
acexmidlemb	|- ( (/) e. B -> ph )

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

Proof of Theorem acexmidlemb

Step	Hyp	Ref	Expression
1		acexmidlem.b	. . . 4 |- B = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ph ) }
2	1	eleq2i 2254	. . 3 |- ( (/) e. B <-> (/) e. { x e. { (/) , { (/) } } | ( x = { (/) } \/ ph ) } )
3		0ex 4142	. . . . 5 |- (/) e. _V
4	3	prid1 3710	. . . 4 |- (/) e. { (/) , { (/) } }
5		eqeq1 2194	. . . . . 6 |- ( x = (/) -> ( x = { (/) } <-> (/) = { (/) } ) )
6	5	orbi1d 792	. . . . 5 |- ( x = (/) -> ( ( x = { (/) } \/ ph ) <-> ( (/) = { (/) } \/ ph ) ) )
7	6	elrab3 2906	. . . 4 |- ( (/) e. { (/) , { (/) } } -> ( (/) e. { x e. { (/) , { (/) } } | ( x = { (/) } \/ ph ) } <-> ( (/) = { (/) } \/ ph ) ) )
8	4, 7	ax-mp 5	. . 3 |- ( (/) e. { x e. { (/) , { (/) } } | ( x = { (/) } \/ ph ) } <-> ( (/) = { (/) } \/ ph ) )
9	2, 8	bitri 184	. 2 |- ( (/) e. B <-> ( (/) = { (/) } \/ ph ) )
10		noel 3438	. . . 4 |- -. (/) e. (/)
11	3	snid 3635	. . . . 5 |- (/) e. { (/) }
12		eleq2 2251	. . . . 5 |- ( (/) = { (/) } -> ( (/) e. (/) <-> (/) e. { (/) } ) )
13	11, 12	mpbiri 168	. . . 4 |- ( (/) = { (/) } -> (/) e. (/) )
14	10, 13	mto 663	. . 3 |- -. (/) = { (/) }
15		orel1 726	. . 3 |- ( -. (/) = { (/) } -> ( ( (/) = { (/) } \/ ph ) -> ph ) )
16	14, 15	ax-mp 5	. 2 |- ( ( (/) = { (/) } \/ ph ) -> ph )
17	9, 16	sylbi 121	1 |- ( (/) e. B -> ph )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   \/ wo 709   = wceq 1363   e. wcel 2158   {crab 2469   (/)c0 3434   {csn 3604   {cpr 3605

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169  ax-nul 4141

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rab 2474  df-v 2751  df-dif 3143  df-un 3145  df-nul 3435  df-sn 3610  df-pr 3611

This theorem is referenced by:  acexmidlem1  5884