Theorem onsucelsucexmidlem1 4560

Description: Lemma for onsucelsucexmid 4562. (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 4156	. . 3 |- (/) e. _V
2	1	prid1 3724	. 2 |- (/) e. { (/) , { (/) } }
3		eqid 2193	. . 3 |- (/) = (/)
4	3	orci 732	. 2 |- ( (/) = (/) \/ ph )
5		eqeq1 2200	. . . 4 |- ( x = (/) -> ( x = (/) <-> (/) = (/) ) )
6	5	orbi1d 792	. . 3 |- ( x = (/) -> ( ( x = (/) \/ ph ) <-> ( (/) = (/) \/ ph ) ) )
7	6	elrab 2916	. 2 |- ( (/) e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } <-> ( (/) e. { (/) , { (/) } } /\ ( (/) = (/) \/ ph ) ) )
8	2, 4, 7	mpbir2an 944	1 |- (/) e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) }

Syntax hints:   \/ wo 709   = wceq 1364   e. wcel 2164   {crab 2476   (/)c0 3446   {csn 3618   {cpr 3619

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-nul 4155

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rab 2481  df-v 2762  df-dif 3155  df-un 3157  df-nul 3447  df-sn 3624  df-pr 3625

This theorem is referenced by:  onsucelsucexmidlem  4561  onsucelsucexmid  4562  acexmidlem2  5915