Theorem regexmidlemm 4509

Description: Lemma for regexmid 4512. 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 4167	. . . 4 |- { (/) } e. _V
2	1	prid2 3683	. . 3 |- { (/) } e. { (/) , { (/) } }
3		eqid 2165	. . . 4 |- { (/) } = { (/) }
4	3	orci 721	. . 3 |- ( { (/) } = { (/) } \/ ( { (/) } = (/) /\ ph ) )
5		eqeq1 2172	. . . . 5 |- ( x = { (/) } -> ( x = { (/) } <-> { (/) } = { (/) } ) )
6		eqeq1 2172	. . . . . 6 |- ( x = { (/) } -> ( x = (/) <-> { (/) } = (/) ) )
7	6	anbi1d 461	. . . . 5 |- ( x = { (/) } -> ( ( x = (/) /\ ph ) <-> ( { (/) } = (/) /\ ph ) ) )
8	5, 7	orbi12d 783	. . . 4 |- ( x = { (/) } -> ( ( x = { (/) } \/ ( x = (/) /\ ph ) ) <-> ( { (/) } = { (/) } \/ ( { (/) } = (/) /\ ph ) ) ) )
9		regexmidlemm.a	. . . 4 |- A = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) }
10	8, 9	elrab2 2885	. . 3 |- ( { (/) } e. A <-> ( { (/) } e. { (/) , { (/) } } /\ ( { (/) } = { (/) } \/ ( { (/) } = (/) /\ ph ) ) ) )
11	2, 4, 10	mpbir2an 932	. 2 |- { (/) } e. A
12		elex2 2742	. 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 698   = wceq 1343   E.wex 1480   e. wcel 2136   {crab 2448   (/)c0 3409   {csn 3576   {cpr 3577

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rab 2453  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583

This theorem is referenced by:  regexmid  4512  reg2exmid  4513  reg3exmid  4557  nnregexmid  4598