MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oteqex Structured version   Visualization version   GIF version

Theorem oteqex 4924
Description: Equivalence of existence implied by equality of ordered triples. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
oteqex (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝑅, 𝑆⟩, 𝑇⟩ → ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) ↔ (𝑅 ∈ V ∧ 𝑆 ∈ V ∧ 𝑇 ∈ V)))

Proof of Theorem oteqex
StepHypRef Expression
1 simp3 1061 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → 𝐶 ∈ V)
21a1i 11 . 2 (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝑅, 𝑆⟩, 𝑇⟩ → ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → 𝐶 ∈ V))
3 simp3 1061 . . 3 ((𝑅 ∈ V ∧ 𝑆 ∈ V ∧ 𝑇 ∈ V) → 𝑇 ∈ V)
4 oteqex2 4923 . . 3 (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝑅, 𝑆⟩, 𝑇⟩ → (𝐶 ∈ V ↔ 𝑇 ∈ V))
53, 4syl5ibr 236 . 2 (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝑅, 𝑆⟩, 𝑇⟩ → ((𝑅 ∈ V ∧ 𝑆 ∈ V ∧ 𝑇 ∈ V) → 𝐶 ∈ V))
6 opex 4893 . . . . . . . 8 𝐴, 𝐵⟩ ∈ V
7 opthg 4906 . . . . . . . 8 ((⟨𝐴, 𝐵⟩ ∈ V ∧ 𝐶 ∈ V) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝑅, 𝑆⟩, 𝑇⟩ ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑅, 𝑆⟩ ∧ 𝐶 = 𝑇)))
86, 7mpan 705 . . . . . . 7 (𝐶 ∈ V → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝑅, 𝑆⟩, 𝑇⟩ ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑅, 𝑆⟩ ∧ 𝐶 = 𝑇)))
98simprbda 652 . . . . . 6 ((𝐶 ∈ V ∧ ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝑅, 𝑆⟩, 𝑇⟩) → ⟨𝐴, 𝐵⟩ = ⟨𝑅, 𝑆⟩)
10 opeqex 4922 . . . . . 6 (⟨𝐴, 𝐵⟩ = ⟨𝑅, 𝑆⟩ → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝑅 ∈ V ∧ 𝑆 ∈ V)))
119, 10syl 17 . . . . 5 ((𝐶 ∈ V ∧ ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝑅, 𝑆⟩, 𝑇⟩) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝑅 ∈ V ∧ 𝑆 ∈ V)))
124adantl 482 . . . . 5 ((𝐶 ∈ V ∧ ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝑅, 𝑆⟩, 𝑇⟩) → (𝐶 ∈ V ↔ 𝑇 ∈ V))
1311, 12anbi12d 746 . . . 4 ((𝐶 ∈ V ∧ ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝑅, 𝑆⟩, 𝑇⟩) → (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐶 ∈ V) ↔ ((𝑅 ∈ V ∧ 𝑆 ∈ V) ∧ 𝑇 ∈ V)))
14 df-3an 1038 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐶 ∈ V))
15 df-3an 1038 . . . 4 ((𝑅 ∈ V ∧ 𝑆 ∈ V ∧ 𝑇 ∈ V) ↔ ((𝑅 ∈ V ∧ 𝑆 ∈ V) ∧ 𝑇 ∈ V))
1613, 14, 153bitr4g 303 . . 3 ((𝐶 ∈ V ∧ ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝑅, 𝑆⟩, 𝑇⟩) → ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) ↔ (𝑅 ∈ V ∧ 𝑆 ∈ V ∧ 𝑇 ∈ V)))
1716expcom 451 . 2 (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝑅, 𝑆⟩, 𝑇⟩ → (𝐶 ∈ V → ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) ↔ (𝑅 ∈ V ∧ 𝑆 ∈ V ∧ 𝑇 ∈ V))))
182, 5, 17pm5.21ndd 369 1 (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝑅, 𝑆⟩, 𝑇⟩ → ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) ↔ (𝑅 ∈ V ∧ 𝑆 ∈ V ∧ 𝑇 ∈ V)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  Vcvv 3186  cop 4154
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator