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Theorem acexmidlema 5828
Description: Lemma for acexmid 5836. (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 2231 . . 3 ({∅} ∈ 𝐴 ↔ {∅} ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)})
3 p0ex 4162 . . . . 5 {∅} ∈ V
43prid2 3678 . . . 4 {∅} ∈ {∅, {∅}}
5 eqeq1 2171 . . . . . 6 (𝑥 = {∅} → (𝑥 = ∅ ↔ {∅} = ∅))
65orbi1d 781 . . . . 5 (𝑥 = {∅} → ((𝑥 = ∅ ∨ 𝜑) ↔ ({∅} = ∅ ∨ 𝜑)))
76elrab3 2879 . . . 4 ({∅} ∈ {∅, {∅}} → ({∅} ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ ({∅} = ∅ ∨ 𝜑)))
84, 7ax-mp 5 . . 3 ({∅} ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ ({∅} = ∅ ∨ 𝜑))
92, 8bitri 183 . 2 ({∅} ∈ 𝐴 ↔ ({∅} = ∅ ∨ 𝜑))
10 noel 3409 . . . 4 ¬ ∅ ∈ ∅
11 0ex 4104 . . . . . 6 ∅ ∈ V
1211snid 3602 . . . . 5 ∅ ∈ {∅}
13 eleq2 2228 . . . . 5 ({∅} = ∅ → (∅ ∈ {∅} ↔ ∅ ∈ ∅))
1412, 13mpbii 147 . . . 4 ({∅} = ∅ → ∅ ∈ ∅)
1510, 14mto 652 . . 3 ¬ {∅} = ∅
16 orel1 715 . . 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 698   = wceq 1342  wcel 2135  {crab 2446  c0 3405  {csn 3571  {cpr 3572
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 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4095  ax-nul 4103  ax-pow 4148
This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-rab 2451  df-v 2724  df-dif 3114  df-un 3116  df-in 3118  df-ss 3125  df-nul 3406  df-pw 3556  df-sn 3577  df-pr 3578
This theorem is referenced by:  acexmidlem1  5833
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