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Theorem acexmidlema 5531
Description: Lemma for acexmid 5539. (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 2120 . . 3 ({∅} ∈ 𝐴 ↔ {∅} ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)})
3 p0ex 3967 . . . . 5 {∅} ∈ V
43prid2 3505 . . . 4 {∅} ∈ {∅, {∅}}
5 eqeq1 2062 . . . . . 6 (𝑥 = {∅} → (𝑥 = ∅ ↔ {∅} = ∅))
65orbi1d 715 . . . . 5 (𝑥 = {∅} → ((𝑥 = ∅ ∨ 𝜑) ↔ ({∅} = ∅ ∨ 𝜑)))
76elrab3 2722 . . . 4 ({∅} ∈ {∅, {∅}} → ({∅} ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ ({∅} = ∅ ∨ 𝜑)))
84, 7ax-mp 7 . . 3 ({∅} ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ ({∅} = ∅ ∨ 𝜑))
92, 8bitri 177 . 2 ({∅} ∈ 𝐴 ↔ ({∅} = ∅ ∨ 𝜑))
10 noel 3256 . . . 4 ¬ ∅ ∈ ∅
11 0ex 3912 . . . . . 6 ∅ ∈ V
1211snid 3430 . . . . 5 ∅ ∈ {∅}
13 eleq2 2117 . . . . 5 ({∅} = ∅ → (∅ ∈ {∅} ↔ ∅ ∈ ∅))
1412, 13mpbii 140 . . . 4 ({∅} = ∅ → ∅ ∈ ∅)
1510, 14mto 598 . . 3 ¬ {∅} = ∅
16 orel1 654 . . 3 (¬ {∅} = ∅ → (({∅} = ∅ ∨ 𝜑) → 𝜑))
1715, 16ax-mp 7 . 2 (({∅} = ∅ ∨ 𝜑) → 𝜑)
189, 17sylbi 118 1 ({∅} ∈ 𝐴𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 102  wo 639   = wceq 1259  wcel 1409  {crab 2327  c0 3252  {csn 3403  {cpr 3404
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-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-nul 3911  ax-pow 3955
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 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410
This theorem is referenced by:  acexmidlem1  5536
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