Theorem onsucelsucexmidlem1 4652

Description: Lemma for onsucelsucexmid 4654. (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 4239	. . 3 |- (/) e. _V
2	1	prid1 3799	. 2 |- (/) e. { (/) , { (/) } }
3		eqid 2234	. . 3 |- (/) = (/)
4	3	orci 739	. 2 |- ( (/) = (/) \/ ph )
5		eqeq1 2241	. . . 4 |- ( x = (/) -> ( x = (/) <-> (/) = (/) ) )
6	5	orbi1d 799	. . 3 |- ( x = (/) -> ( ( x = (/) \/ ph ) <-> ( (/) = (/) \/ ph ) ) )
7	6	elrab 2975	. 2 |- ( (/) e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } <-> ( (/) e. { (/) , { (/) } } /\ ( (/) = (/) \/ ph ) ) )
8	2, 4, 7	mpbir2an 951	1 |- (/) e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) }

Syntax hints:   \/ wo 716   = wceq 1398   e. wcel 2205   {crab 2526   (/)c0 3510   {csn 3691   {cpr 3692

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-ext 2216  ax-nul 4238

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 3215  df-un 3217  df-nul 3511  df-sn 3697  df-pr 3698

This theorem is referenced by:  onsucelsucexmidlem  4653  onsucelsucexmid  4654  acexmidlem2  6049