Theorem onsucelsucexmidlem1 4576

Description: Lemma for onsucelsucexmid 4578. (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 4171	. . 3 |- (/) e. _V
2	1	prid1 3739	. 2 |- (/) e. { (/) , { (/) } }
3		eqid 2205	. . 3 |- (/) = (/)
4	3	orci 733	. 2 |- ( (/) = (/) \/ ph )
5		eqeq1 2212	. . . 4 |- ( x = (/) -> ( x = (/) <-> (/) = (/) ) )
6	5	orbi1d 793	. . 3 |- ( x = (/) -> ( ( x = (/) \/ ph ) <-> ( (/) = (/) \/ ph ) ) )
7	6	elrab 2929	. 2 |- ( (/) e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } <-> ( (/) e. { (/) , { (/) } } /\ ( (/) = (/) \/ ph ) ) )
8	2, 4, 7	mpbir2an 945	1 |- (/) e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) }

Syntax hints:   \/ wo 710   = wceq 1373   e. wcel 2176   {crab 2488   (/)c0 3460   {csn 3633   {cpr 3634

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187  ax-nul 4170

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rab 2493  df-v 2774  df-dif 3168  df-un 3170  df-nul 3461  df-sn 3639  df-pr 3640

This theorem is referenced by:  onsucelsucexmidlem  4577  onsucelsucexmid  4578  acexmidlem2  5941