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Theorem acexmidlema 5731
Description: Lemma for acexmid 5739. (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 2182 . . 3 ({∅} ∈ 𝐴 ↔ {∅} ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)})
3 p0ex 4080 . . . . 5 {∅} ∈ V
43prid2 3598 . . . 4 {∅} ∈ {∅, {∅}}
5 eqeq1 2122 . . . . . 6 (𝑥 = {∅} → (𝑥 = ∅ ↔ {∅} = ∅))
65orbi1d 763 . . . . 5 (𝑥 = {∅} → ((𝑥 = ∅ ∨ 𝜑) ↔ ({∅} = ∅ ∨ 𝜑)))
76elrab3 2812 . . . 4 ({∅} ∈ {∅, {∅}} → ({∅} ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ ({∅} = ∅ ∨ 𝜑)))
84, 7ax-mp 5 . . 3 ({∅} ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ ({∅} = ∅ ∨ 𝜑))
92, 8bitri 183 . 2 ({∅} ∈ 𝐴 ↔ ({∅} = ∅ ∨ 𝜑))
10 noel 3335 . . . 4 ¬ ∅ ∈ ∅
11 0ex 4023 . . . . . 6 ∅ ∈ V
1211snid 3524 . . . . 5 ∅ ∈ {∅}
13 eleq2 2179 . . . . 5 ({∅} = ∅ → (∅ ∈ {∅} ↔ ∅ ∈ ∅))
1412, 13mpbii 147 . . . 4 ({∅} = ∅ → ∅ ∈ ∅)
1510, 14mto 634 . . 3 ¬ {∅} = ∅
16 orel1 697 . . 3 (¬ {∅} = ∅ → (({∅} = ∅ ∨ 𝜑) → 𝜑))
1715, 16ax-mp 5 . 2 (({∅} = ∅ ∨ 𝜑) → 𝜑)
189, 17sylbi 120 1 ({∅} ∈ 𝐴𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 104  wo 680   = wceq 1314  wcel 1463  {crab 2395  c0 3331  {csn 3495  {cpr 3496
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-rab 2400  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502
This theorem is referenced by:  acexmidlem1  5736
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