Theorem onsucelsucexmidlem1 4381

Description: Lemma for onsucelsucexmid 4383. (Contributed by Jim Kingdon, 2-Aug-2019.)

Assertion

Ref	Expression
onsucelsucexmidlem1	|- (/) e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) }

Distinct variable group:   ph, x

Proof of Theorem onsucelsucexmidlem1

Step	Hyp	Ref	Expression
1		0ex 3995	. . 3 |- (/) e. _V
2	1	prid1 3576	. 2 |- (/) e. { (/) , { (/) } }
3		eqid 2100	. . 3 |- (/) = (/)
4	3	orci 691	. 2 |- ( (/) = (/) \/ ph )
5		eqeq1 2106	. . . 4 |- ( x = (/) -> ( x = (/) <-> (/) = (/) ) )
6	5	orbi1d 746	. . 3 |- ( x = (/) -> ( ( x = (/) \/ ph ) <-> ( (/) = (/) \/ ph ) ) )
7	6	elrab 2793	. 2 |- ( (/) e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } <-> ( (/) e. { (/) , { (/) } } /\ ( (/) = (/) \/ ph ) ) )
8	2, 4, 7	mpbir2an 894	1 |- (/) e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) }

Syntax hints:   \/ wo 670   = wceq 1299   e. wcel 1448   {crab 2379   (/)c0 3310   {csn 3474   {cpr 3475

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-nul 3994

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-rab 2384  df-v 2643  df-dif 3023  df-un 3025  df-nul 3311  df-sn 3480  df-pr 3481

This theorem is referenced by:  onsucelsucexmidlem  4382  onsucelsucexmid  4383  acexmidlem2  5703