Theorem regexmidlemm 4516

Description: Lemma for regexmid 4519. A is inhabited. (Contributed by Jim Kingdon, 3-Sep-2019.)

Hypothesis

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

Assertion

Ref	Expression
regexmidlemm	|- E. y y e. A

Distinct variable groups:   y, A   ph, x, y

Allowed substitution hint:   A( x)

Proof of Theorem regexmidlemm

Step	Hyp	Ref	Expression
1		p0ex 4174	. . . 4 |- { (/) } e. _V
2	1	prid2 3690	. . 3 |- { (/) } e. { (/) , { (/) } }
3		eqid 2170	. . . 4 |- { (/) } = { (/) }
4	3	orci 726	. . 3 |- ( { (/) } = { (/) } \/ ( { (/) } = (/) /\ ph ) )
5		eqeq1 2177	. . . . 5 |- ( x = { (/) } -> ( x = { (/) } <-> { (/) } = { (/) } ) )
6		eqeq1 2177	. . . . . 6 |- ( x = { (/) } -> ( x = (/) <-> { (/) } = (/) ) )
7	6	anbi1d 462	. . . . 5 |- ( x = { (/) } -> ( ( x = (/) /\ ph ) <-> ( { (/) } = (/) /\ ph ) ) )
8	5, 7	orbi12d 788	. . . 4 |- ( x = { (/) } -> ( ( x = { (/) } \/ ( x = (/) /\ ph ) ) <-> ( { (/) } = { (/) } \/ ( { (/) } = (/) /\ ph ) ) ) )
9		regexmidlemm.a	. . . 4 |- A = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) }
10	8, 9	elrab2 2889	. . 3 |- ( { (/) } e. A <-> ( { (/) } e. { (/) , { (/) } } /\ ( { (/) } = { (/) } \/ ( { (/) } = (/) /\ ph ) ) ) )
11	2, 4, 10	mpbir2an 937	. 2 |- { (/) } e. A
12		elex2 2746	. 2 |- ( { (/) } e. A -> E. y y e. A )
13	11, 12	ax-mp 5	1 |- E. y y e. A

Syntax hints:   /\ wa 103   \/ wo 703   = wceq 1348   E.wex 1485   e. wcel 2141   {crab 2452   (/)c0 3414   {csn 3583   {cpr 3584

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590

This theorem is referenced by:  regexmid  4519  reg2exmid  4520  reg3exmid  4564  nnregexmid  4605