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Theorem acexmidlemb 5860
Description: Lemma for acexmid 5867. (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
acexmidlemb (∅ ∈ 𝐵𝜑)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝜑,𝑥

Proof of Theorem acexmidlemb
StepHypRef Expression
1 acexmidlem.b . . . 4 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}
21eleq2i 2244 . . 3 (∅ ∈ 𝐵 ↔ ∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)})
3 0ex 4127 . . . . 5 ∅ ∈ V
43prid1 3697 . . . 4 ∅ ∈ {∅, {∅}}
5 eqeq1 2184 . . . . . 6 (𝑥 = ∅ → (𝑥 = {∅} ↔ ∅ = {∅}))
65orbi1d 791 . . . . 5 (𝑥 = ∅ → ((𝑥 = {∅} ∨ 𝜑) ↔ (∅ = {∅} ∨ 𝜑)))
76elrab3 2894 . . . 4 (∅ ∈ {∅, {∅}} → (∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)} ↔ (∅ = {∅} ∨ 𝜑)))
84, 7ax-mp 5 . . 3 (∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)} ↔ (∅ = {∅} ∨ 𝜑))
92, 8bitri 184 . 2 (∅ ∈ 𝐵 ↔ (∅ = {∅} ∨ 𝜑))
10 noel 3426 . . . 4 ¬ ∅ ∈ ∅
113snid 3622 . . . . 5 ∅ ∈ {∅}
12 eleq2 2241 . . . . 5 (∅ = {∅} → (∅ ∈ ∅ ↔ ∅ ∈ {∅}))
1311, 12mpbiri 168 . . . 4 (∅ = {∅} → ∅ ∈ ∅)
1410, 13mto 662 . . 3 ¬ ∅ = {∅}
15 orel1 725 . . 3 (¬ ∅ = {∅} → ((∅ = {∅} ∨ 𝜑) → 𝜑))
1614, 15ax-mp 5 . 2 ((∅ = {∅} ∨ 𝜑) → 𝜑)
179, 16sylbi 121 1 (∅ ∈ 𝐵𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 105  wo 708   = wceq 1353  wcel 2148  {crab 2459  c0 3422  {csn 3591  {cpr 3592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-nul 4126
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-nul 3423  df-sn 3597  df-pr 3598
This theorem is referenced by:  acexmidlem1  5864
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