Theorem acexmidlemb 5535

Description: Lemma for acexmid 5542. (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 2146	. . 3 |- ( (/) e. B <-> (/) e. { x e. { (/) , { (/) } } | ( x = { (/) } \/ ph ) } )
3		0ex 3913	. . . . 5 |- (/) e. _V
4	3	prid1 3506	. . . 4 |- (/) e. { (/) , { (/) } }
5		eqeq1 2088	. . . . . 6 |- ( x = (/) -> ( x = { (/) } <-> (/) = { (/) } ) )
6	5	orbi1d 738	. . . . 5 |- ( x = (/) -> ( ( x = { (/) } \/ ph ) <-> ( (/) = { (/) } \/ ph ) ) )
7	6	elrab3 2751	. . . 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 182	. 2 |- ( (/) e. B <-> ( (/) = { (/) } \/ ph ) )
10		noel 3262	. . . 4 |- -. (/) e. (/)
11	3	snid 3433	. . . . 5 |- (/) e. { (/) }
12		eleq2 2143	. . . . 5 |- ( (/) = { (/) } -> ( (/) e. (/) <-> (/) e. { (/) } ) )
13	11, 12	mpbiri 166	. . . 4 |- ( (/) = { (/) } -> (/) e. (/) )
14	10, 13	mto 621	. . 3 |- -. (/) = { (/) }
15		orel1 677	. . 3 |- ( -. (/) = { (/) } -> ( ( (/) = { (/) } \/ ph ) -> ph ) )
16	14, 15	ax-mp 7	. 2 |- ( ( (/) = { (/) } \/ ph ) -> ph )
17	9, 16	sylbi 119	1 |- ( (/) e. B -> ph )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 103   \/ wo 662   = wceq 1285   e. wcel 1434   {crab 2353   (/)c0 3258   {csn 3406   {cpr 3407

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-nul 3912

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rab 2358  df-v 2604  df-dif 2976  df-un 2978  df-nul 3259  df-sn 3412  df-pr 3413

This theorem is referenced by:  acexmidlem1  5539