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Theorem acexmidlemab 5768
Description: Lemma for acexmid 5773. (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 (((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∧ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅}) → ¬ 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 3367 . . . 4 ¬ ∅ ∈ ∅
2 0ex 4055 . . . . . 6 ∅ ∈ V
32snid 3556 . . . . 5 ∅ ∈ {∅}
4 eleq2 2203 . . . . 5 (∅ = {∅} → (∅ ∈ ∅ ↔ ∅ ∈ {∅}))
53, 4mpbiri 167 . . . 4 (∅ = {∅} → ∅ ∈ ∅)
61, 5mto 651 . . 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 5767 . . . . . . . . 9 (𝜑𝐴 = 𝐵)
11 id 19 . . . . . . . . . 10 (𝐴 = 𝐵𝐴 = 𝐵)
12 eleq1 2202 . . . . . . . . . . . 12 (𝐴 = 𝐵 → (𝐴𝑢𝐵𝑢))
1312anbi1d 460 . . . . . . . . . . 11 (𝐴 = 𝐵 → ((𝐴𝑢𝑣𝑢) ↔ (𝐵𝑢𝑣𝑢)))
1413rexbidv 2438 . . . . . . . . . 10 (𝐴 = 𝐵 → (∃𝑢𝑦 (𝐴𝑢𝑣𝑢) ↔ ∃𝑢𝑦 (𝐵𝑢𝑣𝑢)))
1511, 14riotaeqbidv 5733 . . . . . . . . 9 (𝐴 = 𝐵 → (𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)))
1610, 15syl 14 . . . . . . . 8 (𝜑 → (𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)))
1716eqeq1d 2148 . . . . . . 7 (𝜑 → ((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ↔ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = ∅))
1817biimpa 294 . . . . . 6 ((𝜑 ∧ (𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅) → (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = ∅)
1918adantrr 470 . . . . 5 ((𝜑 ∧ ((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∧ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅})) → (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = ∅)
20 simprr 521 . . . . 5 ((𝜑 ∧ ((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∧ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅})) → (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅})
2119, 20eqtr3d 2174 . . . 4 ((𝜑 ∧ ((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∧ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅})) → ∅ = {∅})
2221ex 114 . . 3 (𝜑 → (((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∧ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅}) → ∅ = {∅}))
236, 22mtoi 653 . 2 (𝜑 → ¬ ((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∧ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅}))
2423con2i 616 1 (((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∧ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅}) → ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wo 697   = wceq 1331  wcel 1480  wrex 2417  {crab 2420  c0 3363  {csn 3527  {cpr 3528  crio 5729
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-nul 4054
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-nul 3364  df-sn 3533  df-uni 3737  df-iota 5088  df-riota 5730
This theorem is referenced by:  acexmidlem1  5770
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