Theorem acexmidlema 6049

Description: Lemma for acexmid 6057. (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 2301	. . 3 |- ( { (/) } e. A <-> { (/) } e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } )
3		p0ex 4306	. . . . 5 |- { (/) } e. _V
4	3	prid2 3803	. . . 4 |- { (/) } e. { (/) , { (/) } }
5		eqeq1 2241	. . . . . 6 |- ( x = { (/) } -> ( x = (/) <-> { (/) } = (/) ) )
6	5	orbi1d 799	. . . . 5 |- ( x = { (/) } -> ( ( x = (/) \/ ph ) <-> ( { (/) } = (/) \/ ph ) ) )
7	6	elrab3 2977	. . . 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. A <-> ( { (/) } = (/) \/ ph ) )
10		noel 3516	. . . 4 |- -. (/) e. (/)
11		0ex 4242	. . . . . 6 |- (/) e. _V
12	11	snid 3725	. . . . 5 |- (/) e. { (/) }
13		eleq2 2298	. . . . 5 |- ( { (/) } = (/) -> ( (/) e. { (/) } <-> (/) e. (/) ) )
14	12, 13	mpbii 148	. . . 4 |- ( { (/) } = (/) -> (/) e. (/) )
15	10, 14	mto 668	. . 3 |- -. { (/) } = (/)
16		orel1 733	. . 3 |- ( -. { (/) } = (/) -> ( ( { (/) } = (/) \/ ph ) -> ph ) )
17	15, 16	ax-mp 5	. 2 |- ( ( { (/) } = (/) \/ ph ) -> ph )
18	9, 17	sylbi 121	1 |- ( { (/) } e. A -> ph )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   \/ wo 716   = wceq 1398   e. wcel 2205   {crab 2526   (/)c0 3512   {csn 3694   {cpr 3695

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rab 2531  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701

This theorem is referenced by:  acexmidlem1  6054