Theorem acexmidlema 5773

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
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 2207	. . 3 |- ( { (/) } e. A <-> { (/) } e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } )
3		p0ex 4120	. . . . 5 |- { (/) } e. _V
4	3	prid2 3638	. . . 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. A <-> ( { (/) } = (/) \/ ph ) )
10		noel 3372	. . . 4 |- -. (/) e. (/)
11		0ex 4063	. . . . . 6 |- (/) e. _V
12	11	snid 3563	. . . . 5 |- (/) e. { (/) }
13		eleq2 2204	. . . . 5 |- ( { (/) } = (/) -> ( (/) e. { (/) } <-> (/) e. (/) ) )
14	12, 13	mpbii 147	. . . 4 |- ( { (/) } = (/) -> (/) e. (/) )
15	10, 14	mto 652	. . 3 |- -. { (/) } = (/)
16		orel1 715	. . 3 |- ( -. { (/) } = (/) -> ( ( { (/) } = (/) \/ ph ) -> ph ) )
17	15, 16	ax-mp 5	. 2 |- ( ( { (/) } = (/) \/ ph ) -> ph )
18	9, 17	sylbi 120	1 |- ( { (/) } e. A -> 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-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-nul 4062  ax-pow 4106

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-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539

This theorem is referenced by:  acexmidlem1  5778