Theorem regexmidlemm 4659

Description: Lemma for regexmid 4662. 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 4306	. . . 4 |- { (/) } e. _V
2	1	prid2 3803	. . 3 |- { (/) } e. { (/) , { (/) } }
3		eqid 2234	. . . 4 |- { (/) } = { (/) }
4	3	orci 739	. . 3 |- ( { (/) } = { (/) } \/ ( { (/) } = (/) /\ ph ) )
5		eqeq1 2241	. . . . 5 |- ( x = { (/) } -> ( x = { (/) } <-> { (/) } = { (/) } ) )
6		eqeq1 2241	. . . . . 6 |- ( x = { (/) } -> ( x = (/) <-> { (/) } = (/) ) )
7	6	anbi1d 465	. . . . 5 |- ( x = { (/) } -> ( ( x = (/) /\ ph ) <-> ( { (/) } = (/) /\ ph ) ) )
8	5, 7	orbi12d 801	. . . 4 |- ( x = { (/) } -> ( ( x = { (/) } \/ ( x = (/) /\ ph ) ) <-> ( { (/) } = { (/) } \/ ( { (/) } = (/) /\ ph ) ) ) )
9		regexmidlemm.a	. . . 4 |- A = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) }
10	8, 9	elrab2 2979	. . 3 |- ( { (/) } e. A <-> ( { (/) } e. { (/) , { (/) } } /\ ( { (/) } = { (/) } \/ ( { (/) } = (/) /\ ph ) ) ) )
11	2, 4, 10	mpbir2an 951	. 2 |- { (/) } e. A
12		elex2 2832	. 2 |- ( { (/) } e. A -> E. y y e. A )
13	11, 12	ax-mp 5	1 |- E. y y e. A

Syntax hints:   /\ wa 104   \/ wo 716   = wceq 1398   E.wex 1541   e. wcel 2205   {crab 2526   (/)c0 3512   {csn 3694   {cpr 3695

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-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292

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 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701

This theorem is referenced by:  regexmid  4662  reg2exmid  4663  reg3exmid  4707  nnregexmid  4748