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Theorem regexmidlemm 4217
Description: Lemma for regexmid 4219. A is inhabited. (Contributed by Jim Kingdon, 3-Sep-2019.)
Hypothesis
Ref Expression
regexmidlemm.a A = {x {∅, {∅}} ∣ (x = {∅} (x = ∅ φ))}
Assertion
Ref Expression
regexmidlemm y y A
Distinct variable groups:   y,A   φ,x,y
Allowed substitution hint:   A(x)

Proof of Theorem regexmidlemm
StepHypRef Expression
1 p0ex 3930 . . . 4 {∅} V
21prid2 3468 . . 3 {∅} {∅, {∅}}
3 eqid 2037 . . . 4 {∅} = {∅}
43orci 649 . . 3 ({∅} = {∅} ({∅} = ∅ φ))
5 eqeq1 2043 . . . . 5 (x = {∅} → (x = {∅} ↔ {∅} = {∅}))
6 eqeq1 2043 . . . . . 6 (x = {∅} → (x = ∅ ↔ {∅} = ∅))
76anbi1d 438 . . . . 5 (x = {∅} → ((x = ∅ φ) ↔ ({∅} = ∅ φ)))
85, 7orbi12d 706 . . . 4 (x = {∅} → ((x = {∅} (x = ∅ φ)) ↔ ({∅} = {∅} ({∅} = ∅ φ))))
9 regexmidlemm.a . . . 4 A = {x {∅, {∅}} ∣ (x = {∅} (x = ∅ φ))}
108, 9elrab2 2694 . . 3 ({∅} A ↔ ({∅} {∅, {∅}} ({∅} = {∅} ({∅} = ∅ φ))))
112, 4, 10mpbir2an 848 . 2 {∅} A
12 elex2 2564 . 2 ({∅} Ay y A)
1311, 12ax-mp 7 1 y y A
Colors of variables: wff set class
Syntax hints:   wa 97   wo 628   = wceq 1242  wex 1378   wcel 1390  {crab 2304  c0 3218  {csn 3367  {cpr 3368
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374
This theorem is referenced by:  regexmid  4219  nnregexmid  4285
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