Theorem acexmidlema 5523

Description: Lemma for acexmid 5531. (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 2145	. . 3 |- ( { (/) } e. A <-> { (/) } e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } )
3		p0ex 3959	. . . . 5 |- { (/) } e. _V
4	3	prid2 3499	. . . 4 |- { (/) } e. { (/) , { (/) } }
5		eqeq1 2087	. . . . . 6 |- ( x = { (/) } -> ( x = (/) <-> { (/) } = (/) ) )
6	5	orbi1d 737	. . . . 5 |- ( x = { (/) } -> ( ( x = (/) \/ ph ) <-> ( { (/) } = (/) \/ ph ) ) )
7	6	elrab3 2750	. . . 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. A <-> ( { (/) } = (/) \/ ph ) )
10		noel 3255	. . . 4 |- -. (/) e. (/)
11		0ex 3905	. . . . . 6 |- (/) e. _V
12	11	snid 3425	. . . . 5 |- (/) e. { (/) }
13		eleq2 2142	. . . . 5 |- ( { (/) } = (/) -> ( (/) e. { (/) } <-> (/) e. (/) ) )
14	12, 13	mpbii 146	. . . 4 |- ( { (/) } = (/) -> (/) e. (/) )
15	10, 14	mto 620	. . 3 |- -. { (/) } = (/)
16		orel1 676	. . 3 |- ( -. { (/) } = (/) -> ( ( { (/) } = (/) \/ ph ) -> ph ) )
17	15, 16	ax-mp 7	. 2 |- ( ( { (/) } = (/) \/ ph ) -> ph )
18	9, 17	sylbi 119	1 |- ( { (/) } e. A -> ph )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 103   \/ wo 661   = wceq 1284   e. wcel 1433   {crab 2352   (/)c0 3251   {csn 3398   {cpr 3399

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-nul 3904  ax-pow 3948

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rab 2357  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405

This theorem is referenced by:  acexmidlem1  5528