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Theorem prneimg 3570
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 3569 . . . . 5 (((𝐴𝑈𝐵𝑉) ∧ (𝐶𝑋𝐷𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
2 orddi 742 . . . . . 6 (((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)) ↔ (((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐴 = 𝐶𝐵 = 𝐶)) ∧ ((𝐵 = 𝐷𝐴 = 𝐷) ∧ (𝐵 = 𝐷𝐵 = 𝐶))))
3 simpll 489 . . . . . . 7 ((((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐴 = 𝐶𝐵 = 𝐶)) ∧ ((𝐵 = 𝐷𝐴 = 𝐷) ∧ (𝐵 = 𝐷𝐵 = 𝐶))) → (𝐴 = 𝐶𝐴 = 𝐷))
4 pm1.4 654 . . . . . . . 8 ((𝐵 = 𝐷𝐵 = 𝐶) → (𝐵 = 𝐶𝐵 = 𝐷))
54ad2antll 468 . . . . . . 7 ((((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐴 = 𝐶𝐵 = 𝐶)) ∧ ((𝐵 = 𝐷𝐴 = 𝐷) ∧ (𝐵 = 𝐷𝐵 = 𝐶))) → (𝐵 = 𝐶𝐵 = 𝐷))
63, 5jca 294 . . . . . 6 ((((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐴 = 𝐶𝐵 = 𝐶)) ∧ ((𝐵 = 𝐷𝐴 = 𝐷) ∧ (𝐵 = 𝐷𝐵 = 𝐶))) → ((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐵 = 𝐶𝐵 = 𝐷)))
72, 6sylbi 118 . . . . 5 (((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)) → ((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐵 = 𝐶𝐵 = 𝐷)))
81, 7syl6bi 156 . . . 4 (((𝐴𝑈𝐵𝑉) ∧ (𝐶𝑋𝐷𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} → ((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐵 = 𝐶𝐵 = 𝐷))))
9 oranim 816 . . . . . 6 ((𝐴 = 𝐶𝐴 = 𝐷) → ¬ (¬ 𝐴 = 𝐶 ∧ ¬ 𝐴 = 𝐷))
10 df-ne 2219 . . . . . . 7 (𝐴𝐶 ↔ ¬ 𝐴 = 𝐶)
11 df-ne 2219 . . . . . . 7 (𝐴𝐷 ↔ ¬ 𝐴 = 𝐷)
1210, 11anbi12i 441 . . . . . 6 ((𝐴𝐶𝐴𝐷) ↔ (¬ 𝐴 = 𝐶 ∧ ¬ 𝐴 = 𝐷))
139, 12sylnibr 610 . . . . 5 ((𝐴 = 𝐶𝐴 = 𝐷) → ¬ (𝐴𝐶𝐴𝐷))
14 oranim 816 . . . . . 6 ((𝐵 = 𝐶𝐵 = 𝐷) → ¬ (¬ 𝐵 = 𝐶 ∧ ¬ 𝐵 = 𝐷))
15 df-ne 2219 . . . . . . 7 (𝐵𝐶 ↔ ¬ 𝐵 = 𝐶)
16 df-ne 2219 . . . . . . 7 (𝐵𝐷 ↔ ¬ 𝐵 = 𝐷)
1715, 16anbi12i 441 . . . . . 6 ((𝐵𝐶𝐵𝐷) ↔ (¬ 𝐵 = 𝐶 ∧ ¬ 𝐵 = 𝐷))
1814, 17sylnibr 610 . . . . 5 ((𝐵 = 𝐶𝐵 = 𝐷) → ¬ (𝐵𝐶𝐵𝐷))
1913, 18anim12i 325 . . . 4 (((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐵 = 𝐶𝐵 = 𝐷)) → (¬ (𝐴𝐶𝐴𝐷) ∧ ¬ (𝐵𝐶𝐵𝐷)))
208, 19syl6 33 . . 3 (((𝐴𝑈𝐵𝑉) ∧ (𝐶𝑋𝐷𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} → (¬ (𝐴𝐶𝐴𝐷) ∧ ¬ (𝐵𝐶𝐵𝐷))))
21 pm4.56 815 . . 3 ((¬ (𝐴𝐶𝐴𝐷) ∧ ¬ (𝐵𝐶𝐵𝐷)) ↔ ¬ ((𝐴𝐶𝐴𝐷) ∨ (𝐵𝐶𝐵𝐷)))
2220, 21syl6ib 154 . 2 (((𝐴𝑈𝐵𝑉) ∧ (𝐶𝑋𝐷𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} → ¬ ((𝐴𝐶𝐴𝐷) ∨ (𝐵𝐶𝐵𝐷))))
2322necon2ad 2275 1 (((𝐴𝑈𝐵𝑉) ∧ (𝐶𝑋𝐷𝑌)) → (((𝐴𝐶𝐴𝐷) ∨ (𝐵𝐶𝐵𝐷)) → {𝐴, 𝐵} ≠ {𝐶, 𝐷}))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wo 637   = wceq 1257  wcel 1407  wne 2218  {cpr 3401
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 552  ax-in2 553  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ne 2219  df-v 2574  df-un 2947  df-sn 3406  df-pr 3407
This theorem is referenced by: (None)
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