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Theorem regexmidlemm 4450
Description: Lemma for regexmid 4453. 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 4115 . . . 4 {∅} ∈ V
21prid2 3633 . . 3 {∅} ∈ {∅, {∅}}
3 eqid 2139 . . . 4 {∅} = {∅}
43orci 720 . . 3 ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑))
5 eqeq1 2146 . . . . 5 (𝑥 = {∅} → (𝑥 = {∅} ↔ {∅} = {∅}))
6 eqeq1 2146 . . . . . 6 (𝑥 = {∅} → (𝑥 = ∅ ↔ {∅} = ∅))
76anbi1d 460 . . . . 5 (𝑥 = {∅} → ((𝑥 = ∅ ∧ 𝜑) ↔ ({∅} = ∅ ∧ 𝜑)))
85, 7orbi12d 782 . . . 4 (𝑥 = {∅} → ((𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑)) ↔ ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑))))
9 regexmidlemm.a . . . 4 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}
108, 9elrab2 2843 . . 3 ({∅} ∈ 𝐴 ↔ ({∅} ∈ {∅, {∅}} ∧ ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑))))
112, 4, 10mpbir2an 926 . 2 {∅} ∈ 𝐴
12 elex2 2702 . 2 ({∅} ∈ 𝐴 → ∃𝑦 𝑦𝐴)
1311, 12ax-mp 5 1 𝑦 𝑦𝐴
Colors of variables: wff set class
Syntax hints:  wa 103  wo 697   = wceq 1331  wex 1468  wcel 1480  {crab 2420  c0 3363  {csn 3527  {cpr 3528
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4049  ax-nul 4057  ax-pow 4101
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534
This theorem is referenced by:  regexmid  4453  reg2exmid  4454  reg3exmid  4497  nnregexmid  4537
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