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Theorem regexmidlemm 4310
Description: Lemma for regexmid 4313. 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 𝑦𝐴
Distinct variable groups:   𝑦,𝐴   𝜑,𝑥,𝑦
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem regexmidlemm
StepHypRef Expression
1 p0ex 3986 . . . 4 {∅} ∈ V
21prid2 3523 . . 3 {∅} ∈ {∅, {∅}}
3 eqid 2083 . . . 4 {∅} = {∅}
43orci 683 . . 3 ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑))
5 eqeq1 2089 . . . . 5 (𝑥 = {∅} → (𝑥 = {∅} ↔ {∅} = {∅}))
6 eqeq1 2089 . . . . . 6 (𝑥 = {∅} → (𝑥 = ∅ ↔ {∅} = ∅))
76anbi1d 453 . . . . 5 (𝑥 = {∅} → ((𝑥 = ∅ ∧ 𝜑) ↔ ({∅} = ∅ ∧ 𝜑)))
85, 7orbi12d 740 . . . 4 (𝑥 = {∅} → ((𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑)) ↔ ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑))))
9 regexmidlemm.a . . . 4 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}
108, 9elrab2 2762 . . 3 ({∅} ∈ 𝐴 ↔ ({∅} ∈ {∅, {∅}} ∧ ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑))))
112, 4, 10mpbir2an 884 . 2 {∅} ∈ 𝐴
12 elex2 2626 . 2 ({∅} ∈ 𝐴 → ∃𝑦 𝑦𝐴)
1311, 12ax-mp 7 1 𝑦 𝑦𝐴
Colors of variables: wff set class
Syntax hints:  wa 102  wo 662   = wceq 1285  wex 1422  wcel 1434  {crab 2357  c0 3269  {csn 3422  {cpr 3423
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-nul 3930  ax-pow 3974
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rab 2362  df-v 2614  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429
This theorem is referenced by:  regexmid  4313  reg2exmid  4314  reg3exmid  4357  nnregexmid  4396
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