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Theorem acexmidlema 5446
Description: Lemma for acexmid 5454. (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 2101 . . 3 ({∅} A ↔ {∅} {x {∅, {∅}} ∣ (x = ∅ φ)})
3 p0ex 3930 . . . . 5 {∅} V
43prid2 3468 . . . 4 {∅} {∅, {∅}}
5 eqeq1 2043 . . . . . 6 (x = {∅} → (x = ∅ ↔ {∅} = ∅))
65orbi1d 704 . . . . 5 (x = {∅} → ((x = ∅ φ) ↔ ({∅} = ∅ φ)))
76elrab3 2693 . . . 4 ({∅} {∅, {∅}} → ({∅} {x {∅, {∅}} ∣ (x = ∅ φ)} ↔ ({∅} = ∅ φ)))
84, 7ax-mp 7 . . 3 ({∅} {x {∅, {∅}} ∣ (x = ∅ φ)} ↔ ({∅} = ∅ φ))
92, 8bitri 173 . 2 ({∅} A ↔ ({∅} = ∅ φ))
10 noel 3222 . . . 4 ¬ ∅
11 0ex 3875 . . . . . 6 V
1211snid 3394 . . . . 5 {∅}
13 eleq2 2098 . . . . 5 ({∅} = ∅ → (∅ {∅} ↔ ∅ ∅))
1412, 13mpbii 136 . . . 4 ({∅} = ∅ → ∅ ∅)
1510, 14mto 587 . . 3 ¬ {∅} = ∅
16 orel1 643 . . 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 628   = wceq 1242   wcel 1390  {crab 2304  c0 3218  {csn 3367  {cpr 3368
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374
This theorem is referenced by:  acexmidlem1  5451
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