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Theorem opthneg 5152
Description: Two ordered pairs are not equal iff their first components or their second components are not equal. (Contributed by AV, 13-Dec-2018.)
Assertion
Ref Expression
opthneg ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩ ↔ (𝐴𝐶𝐵𝐷)))

Proof of Theorem opthneg
StepHypRef Expression
1 df-ne 2990 . 2 (⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩ ↔ ¬ ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
2 opthg 5148 . . . 4 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
32notbid 309 . . 3 ((𝐴𝑉𝐵𝑊) → (¬ ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ ¬ (𝐴 = 𝐶𝐵 = 𝐷)))
4 ianor 995 . . . 4 (¬ (𝐴 = 𝐶𝐵 = 𝐷) ↔ (¬ 𝐴 = 𝐶 ∨ ¬ 𝐵 = 𝐷))
5 df-ne 2990 . . . . 5 (𝐴𝐶 ↔ ¬ 𝐴 = 𝐶)
6 df-ne 2990 . . . . 5 (𝐵𝐷 ↔ ¬ 𝐵 = 𝐷)
75, 6orbi12i 929 . . . 4 ((𝐴𝐶𝐵𝐷) ↔ (¬ 𝐴 = 𝐶 ∨ ¬ 𝐵 = 𝐷))
84, 7bitr4i 269 . . 3 (¬ (𝐴 = 𝐶𝐵 = 𝐷) ↔ (𝐴𝐶𝐵𝐷))
93, 8syl6bb 278 . 2 ((𝐴𝑉𝐵𝑊) → (¬ ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴𝐶𝐵𝐷)))
101, 9syl5bb 274 1 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩ ↔ (𝐴𝐶𝐵𝐷)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 865   = wceq 1637  wcel 2157  wne 2989  cop 4387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-sep 4988  ax-nul 4996  ax-pr 5109
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-rab 3116  df-v 3404  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-sn 4382  df-pr 4384  df-op 4388
This theorem is referenced by:  opthne  5153  zlmodzxznm  42871
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