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Theorem prneimg 4532
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 4530 . . . . 5 (((𝐴𝑈𝐵𝑉) ∧ (𝐶𝑋𝐷𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
2 orddi 949 . . . . . 6 (((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)) ↔ (((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐴 = 𝐶𝐵 = 𝐶)) ∧ ((𝐵 = 𝐷𝐴 = 𝐷) ∧ (𝐵 = 𝐷𝐵 = 𝐶))))
3 simpll 807 . . . . . . 7 ((((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐴 = 𝐶𝐵 = 𝐶)) ∧ ((𝐵 = 𝐷𝐴 = 𝐷) ∧ (𝐵 = 𝐷𝐵 = 𝐶))) → (𝐴 = 𝐶𝐴 = 𝐷))
4 pm1.4 400 . . . . . . . 8 ((𝐵 = 𝐷𝐵 = 𝐶) → (𝐵 = 𝐶𝐵 = 𝐷))
54ad2antll 767 . . . . . . 7 ((((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐴 = 𝐶𝐵 = 𝐶)) ∧ ((𝐵 = 𝐷𝐴 = 𝐷) ∧ (𝐵 = 𝐷𝐵 = 𝐶))) → (𝐵 = 𝐶𝐵 = 𝐷))
63, 5jca 555 . . . . . 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 510 . . . . . 6 (¬ (𝐴𝐶𝐴𝐷) ↔ (¬ 𝐴𝐶 ∨ ¬ 𝐴𝐷))
10 nne 2936 . . . . . . 7 𝐴𝐶𝐴 = 𝐶)
11 nne 2936 . . . . . . 7 𝐴𝐷𝐴 = 𝐷)
1210, 11orbi12i 544 . . . . . 6 ((¬ 𝐴𝐶 ∨ ¬ 𝐴𝐷) ↔ (𝐴 = 𝐶𝐴 = 𝐷))
139, 12bitr2i 265 . . . . 5 ((𝐴 = 𝐶𝐴 = 𝐷) ↔ ¬ (𝐴𝐶𝐴𝐷))
14 ianor 510 . . . . . 6 (¬ (𝐵𝐶𝐵𝐷) ↔ (¬ 𝐵𝐶 ∨ ¬ 𝐵𝐷))
15 nne 2936 . . . . . . 7 𝐵𝐶𝐵 = 𝐶)
16 nne 2936 . . . . . . 7 𝐵𝐷𝐵 = 𝐷)
1715, 16orbi12i 544 . . . . . 6 ((¬ 𝐵𝐶 ∨ ¬ 𝐵𝐷) ↔ (𝐵 = 𝐶𝐵 = 𝐷))
1814, 17bitr2i 265 . . . . 5 ((𝐵 = 𝐶𝐵 = 𝐷) ↔ ¬ (𝐵𝐶𝐵𝐷))
1913, 18anbi12i 735 . . . 4 (((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐵 = 𝐶𝐵 = 𝐷)) ↔ (¬ (𝐴𝐶𝐴𝐷) ∧ ¬ (𝐵𝐶𝐵𝐷)))
208, 19syl6ib 241 . . 3 (((𝐴𝑈𝐵𝑉) ∧ (𝐶𝑋𝐷𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} → (¬ (𝐴𝐶𝐴𝐷) ∧ ¬ (𝐵𝐶𝐵𝐷))))
21 pm4.56 517 . . 3 ((¬ (𝐴𝐶𝐴𝐷) ∧ ¬ (𝐵𝐶𝐵𝐷)) ↔ ¬ ((𝐴𝐶𝐴𝐷) ∨ (𝐵𝐶𝐵𝐷)))
2220, 21syl6ib 241 . 2 (((𝐴𝑈𝐵𝑉) ∧ (𝐶𝑋𝐷𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} → ¬ ((𝐴𝐶𝐴𝐷) ∨ (𝐵𝐶𝐵𝐷))))
2322necon2ad 2947 1 (((𝐴𝑈𝐵𝑉) ∧ (𝐶𝑋𝐷𝑌)) → (((𝐴𝐶𝐴𝐷) ∨ (𝐵𝐶𝐵𝐷)) → {𝐴, 𝐵} ≠ {𝐶, 𝐷}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 382  wa 383   = wceq 1632  wcel 2139  wne 2932  {cpr 4323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-v 3342  df-un 3720  df-sn 4322  df-pr 4324
This theorem is referenced by:  prnebg  4533  opthhausdorff  5127  symg2bas  18038  m2detleib  20659  umgrvad2edg  26325  usgrexmpldifpr  26370  zlmodzxzldeplem  42815
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