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

Theorem acexmidlemab 5449
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
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 3222 . . . 4 ¬ ∅
2 0ex 3875 . . . . . 6 V
32snid 3394 . . . . 5 {∅}
4 eleq2 2098 . . . . 5 (∅ = {∅} → (∅ ∅ ↔ ∅ {∅}))
53, 4mpbiri 157 . . . 4 (∅ = {∅} → ∅ ∅)
61, 5mto 587 . . 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 5448 . . . . . . . . 9 (φA = B)
11 id 19 . . . . . . . . . 10 (A = BA = B)
12 eleq1 2097 . . . . . . . . . . . 12 (A = B → (A uB u))
1312anbi1d 438 . . . . . . . . . . 11 (A = B → ((A u v u) ↔ (B u v u)))
1413rexbidv 2321 . . . . . . . . . 10 (A = B → (u y (A u v u) ↔ u y (B u v u)))
1511, 14riotaeqbidv 5414 . . . . . . . . 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 2045 . . . . . . 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 448 . . . . 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 484 . . . . 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 2071 . . . 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 589 . 2 (φ → ¬ ((v A u y (A u v u)) = ∅ (v B u y (B u v u)) = {∅}))
2423con2i 557 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 628   = wceq 1242   wcel 1390  wrex 2301  {crab 2304  c0 3218  {csn 3367  {cpr 3368  crio 5410
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-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-nul 3874
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-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-nul 3219  df-sn 3373  df-uni 3572  df-iota 4810  df-riota 5411
This theorem is referenced by:  acexmidlem1  5451
  Copyright terms: Public domain W3C validator