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Theorem acexmidlema 5625
Description: Lemma for acexmid 5633. (Contributed by Jim Kingdon, 6-Aug-2019.)
Hypotheses
Ref Expression
acexmidlem.a 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
acexmidlem.b 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}
acexmidlem.c 𝐶 = {𝐴, 𝐵}
Assertion
Ref Expression
acexmidlema ({∅} ∈ 𝐴𝜑)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝜑,𝑥

Proof of Theorem acexmidlema
StepHypRef Expression
1 acexmidlem.a . . . 4 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
21eleq2i 2154 . . 3 ({∅} ∈ 𝐴 ↔ {∅} ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)})
3 p0ex 4014 . . . . 5 {∅} ∈ V
43prid2 3544 . . . 4 {∅} ∈ {∅, {∅}}
5 eqeq1 2094 . . . . . 6 (𝑥 = {∅} → (𝑥 = ∅ ↔ {∅} = ∅))
65orbi1d 740 . . . . 5 (𝑥 = {∅} → ((𝑥 = ∅ ∨ 𝜑) ↔ ({∅} = ∅ ∨ 𝜑)))
76elrab3 2770 . . . 4 ({∅} ∈ {∅, {∅}} → ({∅} ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ ({∅} = ∅ ∨ 𝜑)))
84, 7ax-mp 7 . . 3 ({∅} ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ ({∅} = ∅ ∨ 𝜑))
92, 8bitri 182 . 2 ({∅} ∈ 𝐴 ↔ ({∅} = ∅ ∨ 𝜑))
10 noel 3288 . . . 4 ¬ ∅ ∈ ∅
11 0ex 3958 . . . . . 6 ∅ ∈ V
1211snid 3470 . . . . 5 ∅ ∈ {∅}
13 eleq2 2151 . . . . 5 ({∅} = ∅ → (∅ ∈ {∅} ↔ ∅ ∈ ∅))
1412, 13mpbii 146 . . . 4 ({∅} = ∅ → ∅ ∈ ∅)
1510, 14mto 623 . . 3 ¬ {∅} = ∅
16 orel1 679 . . 3 (¬ {∅} = ∅ → (({∅} = ∅ ∨ 𝜑) → 𝜑))
1715, 16ax-mp 7 . 2 (({∅} = ∅ ∨ 𝜑) → 𝜑)
189, 17sylbi 119 1 ({∅} ∈ 𝐴𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 103  wo 664   = wceq 1289  wcel 1438  {crab 2363  c0 3284  {csn 3441  {cpr 3442
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-nul 3957  ax-pow 4001
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rab 2368  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448
This theorem is referenced by:  acexmidlem1  5630
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