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Theorem acexmidlemb 5504
 Description: Lemma for acexmid 5511. (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 2104 . . 3 (∅ ∈ 𝐵 ↔ ∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)})
3 0ex 3884 . . . . 5 ∅ ∈ V
43prid1 3476 . . . 4 ∅ ∈ {∅, {∅}}
5 eqeq1 2046 . . . . . 6 (𝑥 = ∅ → (𝑥 = {∅} ↔ ∅ = {∅}))
65orbi1d 705 . . . . 5 (𝑥 = ∅ → ((𝑥 = {∅} ∨ 𝜑) ↔ (∅ = {∅} ∨ 𝜑)))
76elrab3 2699 . . . 4 (∅ ∈ {∅, {∅}} → (∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)} ↔ (∅ = {∅} ∨ 𝜑)))
84, 7ax-mp 7 . . 3 (∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)} ↔ (∅ = {∅} ∨ 𝜑))
92, 8bitri 173 . 2 (∅ ∈ 𝐵 ↔ (∅ = {∅} ∨ 𝜑))
10 noel 3228 . . . 4 ¬ ∅ ∈ ∅
113snid 3402 . . . . 5 ∅ ∈ {∅}
12 eleq2 2101 . . . . 5 (∅ = {∅} → (∅ ∈ ∅ ↔ ∅ ∈ {∅}))
1311, 12mpbiri 157 . . . 4 (∅ = {∅} → ∅ ∈ ∅)
1410, 13mto 588 . . 3 ¬ ∅ = {∅}
15 orel1 644 . . 3 (¬ ∅ = {∅} → ((∅ = {∅} ∨ 𝜑) → 𝜑))
1614, 15ax-mp 7 . 2 ((∅ = {∅} ∨ 𝜑) → 𝜑)
179, 16sylbi 114 1 (∅ ∈ 𝐵𝜑)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98   ∨ wo 629   = wceq 1243   ∈ wcel 1393  {crab 2310  ∅c0 3224  {csn 3375  {cpr 3376 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-nul 3883 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-nul 3225  df-sn 3381  df-pr 3382 This theorem is referenced by:  acexmidlem1  5508
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