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Theorem prneimg 4356
Description: Two pairs are not equal if at least one element of the first pair is not contained in the second pair. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
Assertion
Ref Expression
prneimg (((𝐴𝑈𝐵𝑉) ∧ (𝐶𝑋𝐷𝑌)) → (((𝐴𝐶𝐴𝐷) ∨ (𝐵𝐶𝐵𝐷)) → {𝐴, 𝐵} ≠ {𝐶, 𝐷}))

Proof of Theorem prneimg
StepHypRef Expression
1 preq12bg 4354 . . . . 5 (((𝐴𝑈𝐵𝑉) ∧ (𝐶𝑋𝐷𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
2 orddi 912 . . . . . 6 (((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)) ↔ (((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐴 = 𝐶𝐵 = 𝐶)) ∧ ((𝐵 = 𝐷𝐴 = 𝐷) ∧ (𝐵 = 𝐷𝐵 = 𝐶))))
3 simpll 789 . . . . . . 7 ((((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐴 = 𝐶𝐵 = 𝐶)) ∧ ((𝐵 = 𝐷𝐴 = 𝐷) ∧ (𝐵 = 𝐷𝐵 = 𝐶))) → (𝐴 = 𝐶𝐴 = 𝐷))
4 pm1.4 401 . . . . . . . 8 ((𝐵 = 𝐷𝐵 = 𝐶) → (𝐵 = 𝐶𝐵 = 𝐷))
54ad2antll 764 . . . . . . 7 ((((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐴 = 𝐶𝐵 = 𝐶)) ∧ ((𝐵 = 𝐷𝐴 = 𝐷) ∧ (𝐵 = 𝐷𝐵 = 𝐶))) → (𝐵 = 𝐶𝐵 = 𝐷))
63, 5jca 554 . . . . . 6 ((((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐴 = 𝐶𝐵 = 𝐶)) ∧ ((𝐵 = 𝐷𝐴 = 𝐷) ∧ (𝐵 = 𝐷𝐵 = 𝐶))) → ((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐵 = 𝐶𝐵 = 𝐷)))
72, 6sylbi 207 . . . . 5 (((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)) → ((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐵 = 𝐶𝐵 = 𝐷)))
81, 7syl6bi 243 . . . 4 (((𝐴𝑈𝐵𝑉) ∧ (𝐶𝑋𝐷𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} → ((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐵 = 𝐶𝐵 = 𝐷))))
9 ianor 509 . . . . . 6 (¬ (𝐴𝐶𝐴𝐷) ↔ (¬ 𝐴𝐶 ∨ ¬ 𝐴𝐷))
10 nne 2794 . . . . . . 7 𝐴𝐶𝐴 = 𝐶)
11 nne 2794 . . . . . . 7 𝐴𝐷𝐴 = 𝐷)
1210, 11orbi12i 543 . . . . . 6 ((¬ 𝐴𝐶 ∨ ¬ 𝐴𝐷) ↔ (𝐴 = 𝐶𝐴 = 𝐷))
139, 12bitr2i 265 . . . . 5 ((𝐴 = 𝐶𝐴 = 𝐷) ↔ ¬ (𝐴𝐶𝐴𝐷))
14 ianor 509 . . . . . 6 (¬ (𝐵𝐶𝐵𝐷) ↔ (¬ 𝐵𝐶 ∨ ¬ 𝐵𝐷))
15 nne 2794 . . . . . . 7 𝐵𝐶𝐵 = 𝐶)
16 nne 2794 . . . . . . 7 𝐵𝐷𝐵 = 𝐷)
1715, 16orbi12i 543 . . . . . 6 ((¬ 𝐵𝐶 ∨ ¬ 𝐵𝐷) ↔ (𝐵 = 𝐶𝐵 = 𝐷))
1814, 17bitr2i 265 . . . . 5 ((𝐵 = 𝐶𝐵 = 𝐷) ↔ ¬ (𝐵𝐶𝐵𝐷))
1913, 18anbi12i 732 . . . 4 (((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐵 = 𝐶𝐵 = 𝐷)) ↔ (¬ (𝐴𝐶𝐴𝐷) ∧ ¬ (𝐵𝐶𝐵𝐷)))
208, 19syl6ib 241 . . 3 (((𝐴𝑈𝐵𝑉) ∧ (𝐶𝑋𝐷𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} → (¬ (𝐴𝐶𝐴𝐷) ∧ ¬ (𝐵𝐶𝐵𝐷))))
21 pm4.56 516 . . 3 ((¬ (𝐴𝐶𝐴𝐷) ∧ ¬ (𝐵𝐶𝐵𝐷)) ↔ ¬ ((𝐴𝐶𝐴𝐷) ∨ (𝐵𝐶𝐵𝐷)))
2220, 21syl6ib 241 . 2 (((𝐴𝑈𝐵𝑉) ∧ (𝐶𝑋𝐷𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} → ¬ ((𝐴𝐶𝐴𝐷) ∨ (𝐵𝐶𝐵𝐷))))
2322necon2ad 2805 1 (((𝐴𝑈𝐵𝑉) ∧ (𝐶𝑋𝐷𝑌)) → (((𝐴𝐶𝐴𝐷) ∨ (𝐵𝐶𝐵𝐷)) → {𝐴, 𝐵} ≠ {𝐶, 𝐷}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 383  wa 384   = wceq 1480  wcel 1987  wne 2790  {cpr 4150
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
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-v 3188  df-un 3560  df-sn 4149  df-pr 4151
This theorem is referenced by:  prnebg  4357  symg2bas  17739  m2detleib  20356  umgrvad2edg  25998  usgrexmpldifpr  26043  zlmodzxzldeplem  41575
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