Theorem acexmidlemb 5774

Description: Lemma for acexmid 5781. (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 2207	. . 3 |- ( (/) e. B <-> (/) e. { x e. { (/) , { (/) } } | ( x = { (/) } \/ ph ) } )
3		0ex 4063	. . . . 5 |- (/) e. _V
4	3	prid1 3637	. . . 4 |- (/) e. { (/) , { (/) } }
5		eqeq1 2147	. . . . . 6 |- ( x = (/) -> ( x = { (/) } <-> (/) = { (/) } ) )
6	5	orbi1d 781	. . . . 5 |- ( x = (/) -> ( ( x = { (/) } \/ ph ) <-> ( (/) = { (/) } \/ ph ) ) )
7	6	elrab3 2845	. . . 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 3372	. . . 4 |- -. (/) e. (/)
11	3	snid 3563	. . . . 5 |- (/) e. { (/) }
12		eleq2 2204	. . . . 5 |- ( (/) = { (/) } -> ( (/) e. (/) <-> (/) e. { (/) } ) )
13	11, 12	mpbiri 167	. . . 4 |- ( (/) = { (/) } -> (/) e. (/) )
14	10, 13	mto 652	. . 3 |- -. (/) = { (/) }
15		orel1 715	. . 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 698   = wceq 1332   e. wcel 1481   {crab 2421   (/)c0 3368   {csn 3532   {cpr 3533

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-nul 4062

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rab 2426  df-v 2691  df-dif 3078  df-un 3080  df-nul 3369  df-sn 3538  df-pr 3539

This theorem is referenced by:  acexmidlem1  5778