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Theorem acexmidlemab 5430
Description: Lemma for acexmid 5435. (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
acexmidlemab (((v A u y (A u v u)) = ∅ (v B u y (B u v u)) = {∅}) → ¬ φ)
Distinct variable groups:   x,y,v,u,A   x,B,y,v,u   x,𝐶,y,v,u   φ,x,y,v,u

Proof of Theorem acexmidlemab
StepHypRef Expression
1 noel 3205 . . . 4 ¬ ∅
2 0ex 3858 . . . . . 6 V
32snid 3377 . . . . 5 {∅}
4 eleq2 2083 . . . . 5 (∅ = {∅} → (∅ ∅ ↔ ∅ {∅}))
53, 4mpbiri 157 . . . 4 (∅ = {∅} → ∅ ∅)
61, 5mto 575 . . 3 ¬ ∅ = {∅}
7 acexmidlem.a . . . . . . . . . 10 A = {x {∅, {∅}} ∣ (x = ∅ φ)}
8 acexmidlem.b . . . . . . . . . 10 B = {x {∅, {∅}} ∣ (x = {∅} φ)}
9 acexmidlem.c . . . . . . . . . 10 𝐶 = {A, B}
107, 8, 9acexmidlemph 5429 . . . . . . . . 9 (φA = B)
11 id 19 . . . . . . . . . 10 (A = BA = B)
12 eleq1 2082 . . . . . . . . . . . 12 (A = B → (A uB u))
1312anbi1d 441 . . . . . . . . . . 11 (A = B → ((A u v u) ↔ (B u v u)))
1413rexbidv 2305 . . . . . . . . . 10 (A = B → (u y (A u v u) ↔ u y (B u v u)))
1511, 14riotaeqbidv 5396 . . . . . . . . 9 (A = B → (v A u y (A u v u)) = (v B u y (B u v u)))
1610, 15syl 14 . . . . . . . 8 (φ → (v A u y (A u v u)) = (v B u y (B u v u)))
1716eqeq1d 2030 . . . . . . 7 (φ → ((v A u y (A u v u)) = ∅ ↔ (v B u y (B u v u)) = ∅))
1817biimpa 280 . . . . . 6 ((φ (v A u y (A u v u)) = ∅) → (v B u y (B u v u)) = ∅)
1918adantrr 451 . . . . 5 ((φ ((v A u y (A u v u)) = ∅ (v B u y (B u v u)) = {∅})) → (v B u y (B u v u)) = ∅)
20 simprr 472 . . . . 5 ((φ ((v A u y (A u v u)) = ∅ (v B u y (B u v u)) = {∅})) → (v B u y (B u v u)) = {∅})
2119, 20eqtr3d 2056 . . . 4 ((φ ((v A u y (A u v u)) = ∅ (v B u y (B u v u)) = {∅})) → ∅ = {∅})
2221ex 108 . . 3 (φ → (((v A u y (A u v u)) = ∅ (v B u y (B u v u)) = {∅}) → ∅ = {∅}))
236, 22mtoi 577 . 2 (φ → ¬ ((v A u y (A u v u)) = ∅ (v B u y (B u v u)) = {∅}))
2423con2i 545 1 (((v A u y (A u v u)) = ∅ (v B u y (B u v u)) = {∅}) → ¬ φ)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   wo 616   = wceq 1228   wcel 1374  wrex 2285  {crab 2288  c0 3201  {csn 3350  {cpr 3351  crio 5392
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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-nul 3857
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-rab 2293  df-v 2537  df-dif 2897  df-nul 3202  df-sn 3356  df-uni 3555  df-iota 4794  df-riota 5393
This theorem is referenced by:  acexmidlem1  5432
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