Theorem regexmidlemm 4533

Description: Lemma for regexmid 4536. 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 4190	. . . 4 |- { (/) } e. _V
2	1	prid2 3701	. . 3 |- { (/) } e. { (/) , { (/) } }
3		eqid 2177	. . . 4 |- { (/) } = { (/) }
4	3	orci 731	. . 3 |- ( { (/) } = { (/) } \/ ( { (/) } = (/) /\ ph ) )
5		eqeq1 2184	. . . . 5 |- ( x = { (/) } -> ( x = { (/) } <-> { (/) } = { (/) } ) )
6		eqeq1 2184	. . . . . 6 |- ( x = { (/) } -> ( x = (/) <-> { (/) } = (/) ) )
7	6	anbi1d 465	. . . . 5 |- ( x = { (/) } -> ( ( x = (/) /\ ph ) <-> ( { (/) } = (/) /\ ph ) ) )
8	5, 7	orbi12d 793	. . . 4 |- ( x = { (/) } -> ( ( x = { (/) } \/ ( x = (/) /\ ph ) ) <-> ( { (/) } = { (/) } \/ ( { (/) } = (/) /\ ph ) ) ) )
9		regexmidlemm.a	. . . 4 |- A = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) }
10	8, 9	elrab2 2898	. . 3 |- ( { (/) } e. A <-> ( { (/) } e. { (/) , { (/) } } /\ ( { (/) } = { (/) } \/ ( { (/) } = (/) /\ ph ) ) ) )
11	2, 4, 10	mpbir2an 942	. 2 |- { (/) } e. A
12		elex2 2755	. 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 708   = wceq 1353   E.wex 1492   e. wcel 2148   {crab 2459   (/)c0 3424   {csn 3594   {cpr 3595

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601

This theorem is referenced by:  regexmid  4536  reg2exmid  4537  reg3exmid  4581  nnregexmid  4622