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Theorem onsucelsucexmidlem1 4280
Description: Lemma for onsucelsucexmid 4282. (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 3911 . . 3 ∅ ∈ V
21prid1 3503 . 2 ∅ ∈ {∅, {∅}}
3 eqid 2056 . . 3 ∅ = ∅
43orci 660 . 2 (∅ = ∅ ∨ 𝜑)
5 eqeq1 2062 . . . 4 (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅))
65orbi1d 715 . . 3 (𝑥 = ∅ → ((𝑥 = ∅ ∨ 𝜑) ↔ (∅ = ∅ ∨ 𝜑)))
76elrab 2720 . 2 (∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ (∅ ∈ {∅, {∅}} ∧ (∅ = ∅ ∨ 𝜑)))
82, 4, 7mpbir2an 860 1 ∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
Colors of variables: wff set class
Syntax hints:  wo 639   = wceq 1259  wcel 1409  {crab 2327  c0 3251  {csn 3402  {cpr 3403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-nul 3910
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rab 2332  df-v 2576  df-dif 2947  df-un 2949  df-nul 3252  df-sn 3408  df-pr 3409
This theorem is referenced by:  onsucelsucexmidlem  4281  onsucelsucexmid  4282  acexmidlem2  5536
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