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Theorem regexmidlemm 4417
Description: Lemma for regexmid 4420. 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 4082 . . . 4 {∅} ∈ V
21prid2 3600 . . 3 {∅} ∈ {∅, {∅}}
3 eqid 2117 . . . 4 {∅} = {∅}
43orci 705 . . 3 ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑))
5 eqeq1 2124 . . . . 5 (𝑥 = {∅} → (𝑥 = {∅} ↔ {∅} = {∅}))
6 eqeq1 2124 . . . . . 6 (𝑥 = {∅} → (𝑥 = ∅ ↔ {∅} = ∅))
76anbi1d 460 . . . . 5 (𝑥 = {∅} → ((𝑥 = ∅ ∧ 𝜑) ↔ ({∅} = ∅ ∧ 𝜑)))
85, 7orbi12d 767 . . . 4 (𝑥 = {∅} → ((𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑)) ↔ ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑))))
9 regexmidlemm.a . . . 4 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}
108, 9elrab2 2816 . . 3 ({∅} ∈ 𝐴 ↔ ({∅} ∈ {∅, {∅}} ∧ ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑))))
112, 4, 10mpbir2an 911 . 2 {∅} ∈ 𝐴
12 elex2 2676 . 2 ({∅} ∈ 𝐴 → ∃𝑦 𝑦𝐴)
1311, 12ax-mp 5 1 𝑦 𝑦𝐴
Colors of variables: wff set class
Syntax hints:  wa 103  wo 682   = wceq 1316  wex 1453  wcel 1465  {crab 2397  c0 3333  {csn 3497  {cpr 3498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rab 2402  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504
This theorem is referenced by:  regexmid  4420  reg2exmid  4421  reg3exmid  4464  nnregexmid  4504
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