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Theorem acexmidlema 5423
Description: Lemma for acexmid 5431. (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 2082 . . 3 ({∅} A ↔ {∅} {x {∅, {∅}} ∣ (x = ∅ φ)})
3 p0ex 3909 . . . . 5 {∅} V
43prid2 3447 . . . 4 {∅} {∅, {∅}}
5 eqeq1 2024 . . . . . 6 (x = {∅} → (x = ∅ ↔ {∅} = ∅))
65orbi1d 692 . . . . 5 (x = {∅} → ((x = ∅ φ) ↔ ({∅} = ∅ φ)))
76elrab3 2672 . . . 4 ({∅} {∅, {∅}} → ({∅} {x {∅, {∅}} ∣ (x = ∅ φ)} ↔ ({∅} = ∅ φ)))
84, 7ax-mp 7 . . 3 ({∅} {x {∅, {∅}} ∣ (x = ∅ φ)} ↔ ({∅} = ∅ φ))
92, 8bitri 173 . 2 ({∅} A ↔ ({∅} = ∅ φ))
10 noel 3201 . . . 4 ¬ ∅
11 0ex 3854 . . . . . 6 V
1211snid 3373 . . . . 5 {∅}
13 eleq2 2079 . . . . 5 ({∅} = ∅ → (∅ {∅} ↔ ∅ ∅))
1412, 13mpbii 136 . . . 4 ({∅} = ∅ → ∅ ∅)
1510, 14mto 575 . . 3 ¬ {∅} = ∅
16 orel1 631 . . 3 (¬ {∅} = ∅ → (({∅} = ∅ φ) → φ))
1715, 16ax-mp 7 . 2 (({∅} = ∅ φ) → φ)
189, 17sylbi 114 1 ({∅} Aφ)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 98   wo 616   = wceq 1226   wcel 1370  {crab 2284  c0 3197  {csn 3346  {cpr 3347
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 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-nul 3853  ax-pow 3897
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-rab 2289  df-v 2533  df-dif 2893  df-un 2895  df-in 2897  df-ss 2904  df-nul 3198  df-pw 3332  df-sn 3352  df-pr 3353
This theorem is referenced by:  acexmidlem1  5428
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