Theorem regexmidlemm 4275

Description: Lemma for regexmid 4278. 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 3959	. . . 4 |- { (/) } e. _V
2	1	prid2 3499	. . 3 |- { (/) } e. { (/) , { (/) } }
3		eqid 2081	. . . 4 |- { (/) } = { (/) }
4	3	orci 682	. . 3 |- ( { (/) } = { (/) } \/ ( { (/) } = (/) /\ ph ) )
5		eqeq1 2087	. . . . 5 |- ( x = { (/) } -> ( x = { (/) } <-> { (/) } = { (/) } ) )
6		eqeq1 2087	. . . . . 6 |- ( x = { (/) } -> ( x = (/) <-> { (/) } = (/) ) )
7	6	anbi1d 452	. . . . 5 |- ( x = { (/) } -> ( ( x = (/) /\ ph ) <-> ( { (/) } = (/) /\ ph ) ) )
8	5, 7	orbi12d 739	. . . 4 |- ( x = { (/) } -> ( ( x = { (/) } \/ ( x = (/) /\ ph ) ) <-> ( { (/) } = { (/) } \/ ( { (/) } = (/) /\ ph ) ) ) )
9		regexmidlemm.a	. . . 4 |- A = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) }
10	8, 9	elrab2 2751	. . 3 |- ( { (/) } e. A <-> ( { (/) } e. { (/) , { (/) } } /\ ( { (/) } = { (/) } \/ ( { (/) } = (/) /\ ph ) ) ) )
11	2, 4, 10	mpbir2an 883	. 2 |- { (/) } e. A
12		elex2 2615	. 2 |- ( { (/) } e. A -> E. y y e. A )
13	11, 12	ax-mp 7	1 |- E. y y e. A

Syntax hints:   /\ wa 102   \/ wo 661   = wceq 1284   E.wex 1421   e. wcel 1433   {crab 2352   (/)c0 3251   {csn 3398   {cpr 3399

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-nul 3904  ax-pow 3948

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rab 2357  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405

This theorem is referenced by:  regexmid  4278  reg2exmid  4279  reg3exmid  4322  nnregexmid  4360