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Theorem prneimg 3536
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 (((A 𝑈 B 𝑉) (𝐶 𝑋 𝐷 𝑌)) → (((A𝐶 A𝐷) (B𝐶 B𝐷)) → {A, B} ≠ {𝐶, 𝐷}))

Proof of Theorem prneimg
StepHypRef Expression
1 preq12bg 3535 . . . . 5 (((A 𝑈 B 𝑉) (𝐶 𝑋 𝐷 𝑌)) → ({A, B} = {𝐶, 𝐷} ↔ ((A = 𝐶 B = 𝐷) (A = 𝐷 B = 𝐶))))
2 orddi 732 . . . . . 6 (((A = 𝐶 B = 𝐷) (A = 𝐷 B = 𝐶)) ↔ (((A = 𝐶 A = 𝐷) (A = 𝐶 B = 𝐶)) ((B = 𝐷 A = 𝐷) (B = 𝐷 B = 𝐶))))
3 simpll 481 . . . . . . 7 ((((A = 𝐶 A = 𝐷) (A = 𝐶 B = 𝐶)) ((B = 𝐷 A = 𝐷) (B = 𝐷 B = 𝐶))) → (A = 𝐶 A = 𝐷))
4 pm1.4 645 . . . . . . . 8 ((B = 𝐷 B = 𝐶) → (B = 𝐶 B = 𝐷))
54ad2antll 460 . . . . . . 7 ((((A = 𝐶 A = 𝐷) (A = 𝐶 B = 𝐶)) ((B = 𝐷 A = 𝐷) (B = 𝐷 B = 𝐶))) → (B = 𝐶 B = 𝐷))
63, 5jca 290 . . . . . 6 ((((A = 𝐶 A = 𝐷) (A = 𝐶 B = 𝐶)) ((B = 𝐷 A = 𝐷) (B = 𝐷 B = 𝐶))) → ((A = 𝐶 A = 𝐷) (B = 𝐶 B = 𝐷)))
72, 6sylbi 114 . . . . 5 (((A = 𝐶 B = 𝐷) (A = 𝐷 B = 𝐶)) → ((A = 𝐶 A = 𝐷) (B = 𝐶 B = 𝐷)))
81, 7syl6bi 152 . . . 4 (((A 𝑈 B 𝑉) (𝐶 𝑋 𝐷 𝑌)) → ({A, B} = {𝐶, 𝐷} → ((A = 𝐶 A = 𝐷) (B = 𝐶 B = 𝐷))))
9 oranim 806 . . . . . 6 ((A = 𝐶 A = 𝐷) → ¬ (¬ A = 𝐶 ¬ A = 𝐷))
10 df-ne 2203 . . . . . . 7 (A𝐶 ↔ ¬ A = 𝐶)
11 df-ne 2203 . . . . . . 7 (A𝐷 ↔ ¬ A = 𝐷)
1210, 11anbi12i 433 . . . . . 6 ((A𝐶 A𝐷) ↔ (¬ A = 𝐶 ¬ A = 𝐷))
139, 12sylnibr 601 . . . . 5 ((A = 𝐶 A = 𝐷) → ¬ (A𝐶 A𝐷))
14 oranim 806 . . . . . 6 ((B = 𝐶 B = 𝐷) → ¬ (¬ B = 𝐶 ¬ B = 𝐷))
15 df-ne 2203 . . . . . . 7 (B𝐶 ↔ ¬ B = 𝐶)
16 df-ne 2203 . . . . . . 7 (B𝐷 ↔ ¬ B = 𝐷)
1715, 16anbi12i 433 . . . . . 6 ((B𝐶 B𝐷) ↔ (¬ B = 𝐶 ¬ B = 𝐷))
1814, 17sylnibr 601 . . . . 5 ((B = 𝐶 B = 𝐷) → ¬ (B𝐶 B𝐷))
1913, 18anim12i 321 . . . 4 (((A = 𝐶 A = 𝐷) (B = 𝐶 B = 𝐷)) → (¬ (A𝐶 A𝐷) ¬ (B𝐶 B𝐷)))
208, 19syl6 29 . . 3 (((A 𝑈 B 𝑉) (𝐶 𝑋 𝐷 𝑌)) → ({A, B} = {𝐶, 𝐷} → (¬ (A𝐶 A𝐷) ¬ (B𝐶 B𝐷))))
21 pm4.56 805 . . 3 ((¬ (A𝐶 A𝐷) ¬ (B𝐶 B𝐷)) ↔ ¬ ((A𝐶 A𝐷) (B𝐶 B𝐷)))
2220, 21syl6ib 150 . 2 (((A 𝑈 B 𝑉) (𝐶 𝑋 𝐷 𝑌)) → ({A, B} = {𝐶, 𝐷} → ¬ ((A𝐶 A𝐷) (B𝐶 B𝐷))))
2322necon2ad 2256 1 (((A 𝑈 B 𝑉) (𝐶 𝑋 𝐷 𝑌)) → (((A𝐶 A𝐷) (B𝐶 B𝐷)) → {A, B} ≠ {𝐶, 𝐷}))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   wo 628   = wceq 1242   wcel 1390  wne 2201  {cpr 3368
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374
This theorem is referenced by: (None)
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