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Theorem onsucelsucexmidlem1 4401
Description: Lemma for onsucelsucexmid 4403. (Contributed by Jim Kingdon, 2-Aug-2019.)
Assertion
Ref Expression
onsucelsucexmidlem1 ∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
Distinct variable group:   𝜑,𝑥

Proof of Theorem onsucelsucexmidlem1
StepHypRef Expression
1 0ex 4013 . . 3 ∅ ∈ V
21prid1 3593 . 2 ∅ ∈ {∅, {∅}}
3 eqid 2113 . . 3 ∅ = ∅
43orci 703 . 2 (∅ = ∅ ∨ 𝜑)
5 eqeq1 2119 . . . 4 (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅))
65orbi1d 763 . . 3 (𝑥 = ∅ → ((𝑥 = ∅ ∨ 𝜑) ↔ (∅ = ∅ ∨ 𝜑)))
76elrab 2807 . 2 (∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ (∅ ∈ {∅, {∅}} ∧ (∅ = ∅ ∨ 𝜑)))
82, 4, 7mpbir2an 907 1 ∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
Colors of variables: wff set class
Syntax hints:  wo 680   = wceq 1312  wcel 1461  {crab 2392  c0 3327  {csn 3491  {cpr 3492
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-nul 4012
This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-rab 2397  df-v 2657  df-dif 3037  df-un 3039  df-nul 3328  df-sn 3497  df-pr 3498
This theorem is referenced by:  onsucelsucexmidlem  4402  onsucelsucexmid  4403  acexmidlem2  5723
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