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Theorem acexmidlemb 5531
Description: Lemma for acexmid 5538. (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 2120 . . 3 (∅ ∈ 𝐵 ↔ ∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)})
3 0ex 3911 . . . . 5 ∅ ∈ V
43prid1 3503 . . . 4 ∅ ∈ {∅, {∅}}
5 eqeq1 2062 . . . . . 6 (𝑥 = ∅ → (𝑥 = {∅} ↔ ∅ = {∅}))
65orbi1d 715 . . . . 5 (𝑥 = ∅ → ((𝑥 = {∅} ∨ 𝜑) ↔ (∅ = {∅} ∨ 𝜑)))
76elrab3 2721 . . . 4 (∅ ∈ {∅, {∅}} → (∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)} ↔ (∅ = {∅} ∨ 𝜑)))
84, 7ax-mp 7 . . 3 (∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)} ↔ (∅ = {∅} ∨ 𝜑))
92, 8bitri 177 . 2 (∅ ∈ 𝐵 ↔ (∅ = {∅} ∨ 𝜑))
10 noel 3255 . . . 4 ¬ ∅ ∈ ∅
113snid 3429 . . . . 5 ∅ ∈ {∅}
12 eleq2 2117 . . . . 5 (∅ = {∅} → (∅ ∈ ∅ ↔ ∅ ∈ {∅}))
1311, 12mpbiri 161 . . . 4 (∅ = {∅} → ∅ ∈ ∅)
1410, 13mto 598 . . 3 ¬ ∅ = {∅}
15 orel1 654 . . 3 (¬ ∅ = {∅} → ((∅ = {∅} ∨ 𝜑) → 𝜑))
1614, 15ax-mp 7 . 2 ((∅ = {∅} ∨ 𝜑) → 𝜑)
179, 16sylbi 118 1 (∅ ∈ 𝐵𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 102  wo 639   = wceq 1259  wcel 1409  {crab 2327  c0 3251  {csn 3402  {cpr 3403
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-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-nul 3910
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 2947  df-un 2949  df-nul 3252  df-sn 3408  df-pr 3409
This theorem is referenced by:  acexmidlem1  5535
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