Theorem acexmidlemb 5766

Description: Lemma for acexmid 5773. (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 2206	. . 3 |- ( (/) e. B <-> (/) e. { x e. { (/) , { (/) } } | ( x = { (/) } \/ ph ) } )
3		0ex 4055	. . . . 5 |- (/) e. _V
4	3	prid1 3629	. . . 4 |- (/) e. { (/) , { (/) } }
5		eqeq1 2146	. . . . . 6 |- ( x = (/) -> ( x = { (/) } <-> (/) = { (/) } ) )
6	5	orbi1d 780	. . . . 5 |- ( x = (/) -> ( ( x = { (/) } \/ ph ) <-> ( (/) = { (/) } \/ ph ) ) )
7	6	elrab3 2841	. . . 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 183	. 2 |- ( (/) e. B <-> ( (/) = { (/) } \/ ph ) )
10		noel 3367	. . . 4 |- -. (/) e. (/)
11	3	snid 3556	. . . . 5 |- (/) e. { (/) }
12		eleq2 2203	. . . . 5 |- ( (/) = { (/) } -> ( (/) e. (/) <-> (/) e. { (/) } ) )
13	11, 12	mpbiri 167	. . . 4 |- ( (/) = { (/) } -> (/) e. (/) )
14	10, 13	mto 651	. . 3 |- -. (/) = { (/) }
15		orel1 714	. . 3 |- ( -. (/) = { (/) } -> ( ( (/) = { (/) } \/ ph ) -> ph ) )
16	14, 15	ax-mp 5	. 2 |- ( ( (/) = { (/) } \/ ph ) -> ph )
17	9, 16	sylbi 120	1 |- ( (/) e. B -> ph )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 104   \/ wo 697   = wceq 1331   e. wcel 1480   {crab 2420   (/)c0 3363   {csn 3527   {cpr 3528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-nul 4054

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-nul 3364  df-sn 3533  df-pr 3534

This theorem is referenced by:  acexmidlem1  5770