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Theorem regexmidlemm 4201
Description: Lemma for regexmid 4203. 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 3913 . . . 4 {∅} V
21prid2 3451 . . 3 {∅} {∅, {∅}}
3 eqid 2022 . . . 4 {∅} = {∅}
43orci 637 . . 3 ({∅} = {∅} ({∅} = ∅ φ))
5 eqeq1 2028 . . . . 5 (x = {∅} → (x = {∅} ↔ {∅} = {∅}))
6 eqeq1 2028 . . . . . 6 (x = {∅} → (x = ∅ ↔ {∅} = ∅))
76anbi1d 441 . . . . 5 (x = {∅} → ((x = ∅ φ) ↔ ({∅} = ∅ φ)))
85, 7orbi12d 694 . . . 4 (x = {∅} → ((x = {∅} (x = ∅ φ)) ↔ ({∅} = {∅} ({∅} = ∅ φ))))
9 regexmidlemm.a . . . 4 A = {x {∅, {∅}} ∣ (x = {∅} (x = ∅ φ))}
108, 9elrab2 2677 . . 3 ({∅} A ↔ ({∅} {∅, {∅}} ({∅} = {∅} ({∅} = ∅ φ))))
112, 4, 10mpbir2an 837 . 2 {∅} A
12 elex2 2547 . 2 ({∅} Ay y A)
1311, 12ax-mp 7 1 y y A
Colors of variables: wff set class
Syntax hints:   wa 97   wo 616   = wceq 1228  wex 1362   wcel 1374  {crab 2288  c0 3201  {csn 3350  {cpr 3351
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-nul 3857  ax-pow 3901
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rab 2293  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357
This theorem is referenced by:  regexmid  4203  nnregexmid  4269
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