Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  acexmidlemab GIF version

Theorem acexmidlemab 5533
 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
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 3255 . . . 4 ¬ ∅ ∈ ∅
2 0ex 3911 . . . . . 6 ∅ ∈ V
32snid 3429 . . . . 5 ∅ ∈ {∅}
4 eleq2 2117 . . . . 5 (∅ = {∅} → (∅ ∈ ∅ ↔ ∅ ∈ {∅}))
53, 4mpbiri 161 . . . 4 (∅ = {∅} → ∅ ∈ ∅)
61, 5mto 598 . . 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 5532 . . . . . . . . 9 (𝜑𝐴 = 𝐵)
11 id 19 . . . . . . . . . 10 (𝐴 = 𝐵𝐴 = 𝐵)
12 eleq1 2116 . . . . . . . . . . . 12 (𝐴 = 𝐵 → (𝐴𝑢𝐵𝑢))
1312anbi1d 446 . . . . . . . . . . 11 (𝐴 = 𝐵 → ((𝐴𝑢𝑣𝑢) ↔ (𝐵𝑢𝑣𝑢)))
1413rexbidv 2344 . . . . . . . . . 10 (𝐴 = 𝐵 → (∃𝑢𝑦 (𝐴𝑢𝑣𝑢) ↔ ∃𝑢𝑦 (𝐵𝑢𝑣𝑢)))
1511, 14riotaeqbidv 5498 . . . . . . . . 9 (𝐴 = 𝐵 → (𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)))
1610, 15syl 14 . . . . . . . 8 (𝜑 → (𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)))
1716eqeq1d 2064 . . . . . . 7 (𝜑 → ((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ↔ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = ∅))
1817biimpa 284 . . . . . 6 ((𝜑 ∧ (𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅) → (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = ∅)
1918adantrr 456 . . . . 5 ((𝜑 ∧ ((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∧ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅})) → (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = ∅)
20 simprr 492 . . . . 5 ((𝜑 ∧ ((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∧ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅})) → (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅})
2119, 20eqtr3d 2090 . . . 4 ((𝜑 ∧ ((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∧ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅})) → ∅ = {∅})
2221ex 112 . . 3 (𝜑 → (((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∧ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅}) → ∅ = {∅}))
236, 22mtoi 600 . 2 (𝜑 → ¬ ((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∧ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅}))
2423con2i 567 1 (((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∧ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅}) → ¬ 𝜑)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 101   ∨ wo 639   = wceq 1259   ∈ wcel 1409  ∃wrex 2324  {crab 2327  ∅c0 3251  {csn 3402  {cpr 3403  ℩crio 5494 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-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-dif 2947  df-nul 3252  df-sn 3408  df-uni 3608  df-iota 4894  df-riota 5495 This theorem is referenced by:  acexmidlem1  5535
 Copyright terms: Public domain W3C validator