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Theorem prnebg 4856
Description: A (proper) pair is not equal to another (maybe improper) pair if and only if an element of the first pair is not contained in the second pair. (Contributed by Alexander van der Vekens, 16-Jan-2018.)
Assertion
Ref Expression
prnebg (((𝐴𝑈𝐵𝑉) ∧ (𝐶𝑋𝐷𝑌) ∧ 𝐴𝐵) → (((𝐴𝐶𝐴𝐷) ∨ (𝐵𝐶𝐵𝐷)) ↔ {𝐴, 𝐵} ≠ {𝐶, 𝐷}))

Proof of Theorem prnebg
StepHypRef Expression
1 prneimg 4855 . . 3 (((𝐴𝑈𝐵𝑉) ∧ (𝐶𝑋𝐷𝑌)) → (((𝐴𝐶𝐴𝐷) ∨ (𝐵𝐶𝐵𝐷)) → {𝐴, 𝐵} ≠ {𝐶, 𝐷}))
213adant3 1132 . 2 (((𝐴𝑈𝐵𝑉) ∧ (𝐶𝑋𝐷𝑌) ∧ 𝐴𝐵) → (((𝐴𝐶𝐴𝐷) ∨ (𝐵𝐶𝐵𝐷)) → {𝐴, 𝐵} ≠ {𝐶, 𝐷}))
3 ioran 982 . . . . 5 (¬ ((𝐴𝐶𝐴𝐷) ∨ (𝐵𝐶𝐵𝐷)) ↔ (¬ (𝐴𝐶𝐴𝐷) ∧ ¬ (𝐵𝐶𝐵𝐷)))
4 ianor 980 . . . . . . 7 (¬ (𝐴𝐶𝐴𝐷) ↔ (¬ 𝐴𝐶 ∨ ¬ 𝐴𝐷))
5 nne 2944 . . . . . . . 8 𝐴𝐶𝐴 = 𝐶)
6 nne 2944 . . . . . . . 8 𝐴𝐷𝐴 = 𝐷)
75, 6orbi12i 913 . . . . . . 7 ((¬ 𝐴𝐶 ∨ ¬ 𝐴𝐷) ↔ (𝐴 = 𝐶𝐴 = 𝐷))
84, 7bitri 274 . . . . . 6 (¬ (𝐴𝐶𝐴𝐷) ↔ (𝐴 = 𝐶𝐴 = 𝐷))
9 ianor 980 . . . . . . 7 (¬ (𝐵𝐶𝐵𝐷) ↔ (¬ 𝐵𝐶 ∨ ¬ 𝐵𝐷))
10 nne 2944 . . . . . . . 8 𝐵𝐶𝐵 = 𝐶)
11 nne 2944 . . . . . . . 8 𝐵𝐷𝐵 = 𝐷)
1210, 11orbi12i 913 . . . . . . 7 ((¬ 𝐵𝐶 ∨ ¬ 𝐵𝐷) ↔ (𝐵 = 𝐶𝐵 = 𝐷))
139, 12bitri 274 . . . . . 6 (¬ (𝐵𝐶𝐵𝐷) ↔ (𝐵 = 𝐶𝐵 = 𝐷))
148, 13anbi12i 627 . . . . 5 ((¬ (𝐴𝐶𝐴𝐷) ∧ ¬ (𝐵𝐶𝐵𝐷)) ↔ ((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐵 = 𝐶𝐵 = 𝐷)))
153, 14bitri 274 . . . 4 (¬ ((𝐴𝐶𝐴𝐷) ∨ (𝐵𝐶𝐵𝐷)) ↔ ((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐵 = 𝐶𝐵 = 𝐷)))
16 anddi 1009 . . . . 5 (((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐵 = 𝐶𝐵 = 𝐷)) ↔ (((𝐴 = 𝐶𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐷)) ∨ ((𝐴 = 𝐷𝐵 = 𝐶) ∨ (𝐴 = 𝐷𝐵 = 𝐷))))
17 eqtr3 2758 . . . . . . . . . 10 ((𝐴 = 𝐶𝐵 = 𝐶) → 𝐴 = 𝐵)
18 eqneqall 2951 . . . . . . . . . 10 (𝐴 = 𝐵 → (𝐴𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷}))
1917, 18syl 17 . . . . . . . . 9 ((𝐴 = 𝐶𝐵 = 𝐶) → (𝐴𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷}))
20 preq12 4739 . . . . . . . . . 10 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})
2120a1d 25 . . . . . . . . 9 ((𝐴 = 𝐶𝐵 = 𝐷) → (𝐴𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷}))
2219, 21jaoi 855 . . . . . . . 8 (((𝐴 = 𝐶𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐷)) → (𝐴𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷}))
23 preq12 4739 . . . . . . . . . . 11 ((𝐴 = 𝐷𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐷, 𝐶})
24 prcom 4736 . . . . . . . . . . 11 {𝐷, 𝐶} = {𝐶, 𝐷}
2523, 24eqtrdi 2788 . . . . . . . . . 10 ((𝐴 = 𝐷𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐶, 𝐷})
2625a1d 25 . . . . . . . . 9 ((𝐴 = 𝐷𝐵 = 𝐶) → (𝐴𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷}))
27 eqtr3 2758 . . . . . . . . . 10 ((𝐴 = 𝐷𝐵 = 𝐷) → 𝐴 = 𝐵)
2827, 18syl 17 . . . . . . . . 9 ((𝐴 = 𝐷𝐵 = 𝐷) → (𝐴𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷}))
2926, 28jaoi 855 . . . . . . . 8 (((𝐴 = 𝐷𝐵 = 𝐶) ∨ (𝐴 = 𝐷𝐵 = 𝐷)) → (𝐴𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷}))
3022, 29jaoi 855 . . . . . . 7 ((((𝐴 = 𝐶𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐷)) ∨ ((𝐴 = 𝐷𝐵 = 𝐶) ∨ (𝐴 = 𝐷𝐵 = 𝐷))) → (𝐴𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷}))
3130com12 32 . . . . . 6 (𝐴𝐵 → ((((𝐴 = 𝐶𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐷)) ∨ ((𝐴 = 𝐷𝐵 = 𝐶) ∨ (𝐴 = 𝐷𝐵 = 𝐷))) → {𝐴, 𝐵} = {𝐶, 𝐷}))
32313ad2ant3 1135 . . . . 5 (((𝐴𝑈𝐵𝑉) ∧ (𝐶𝑋𝐷𝑌) ∧ 𝐴𝐵) → ((((𝐴 = 𝐶𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐷)) ∨ ((𝐴 = 𝐷𝐵 = 𝐶) ∨ (𝐴 = 𝐷𝐵 = 𝐷))) → {𝐴, 𝐵} = {𝐶, 𝐷}))
3316, 32biimtrid 241 . . . 4 (((𝐴𝑈𝐵𝑉) ∧ (𝐶𝑋𝐷𝑌) ∧ 𝐴𝐵) → (((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐵 = 𝐶𝐵 = 𝐷)) → {𝐴, 𝐵} = {𝐶, 𝐷}))
3415, 33biimtrid 241 . . 3 (((𝐴𝑈𝐵𝑉) ∧ (𝐶𝑋𝐷𝑌) ∧ 𝐴𝐵) → (¬ ((𝐴𝐶𝐴𝐷) ∨ (𝐵𝐶𝐵𝐷)) → {𝐴, 𝐵} = {𝐶, 𝐷}))
3534necon1ad 2957 . 2 (((𝐴𝑈𝐵𝑉) ∧ (𝐶𝑋𝐷𝑌) ∧ 𝐴𝐵) → ({𝐴, 𝐵} ≠ {𝐶, 𝐷} → ((𝐴𝐶𝐴𝐷) ∨ (𝐵𝐶𝐵𝐷))))
362, 35impbid 211 1 (((𝐴𝑈𝐵𝑉) ∧ (𝐶𝑋𝐷𝑌) ∧ 𝐴𝐵) → (((𝐴𝐶𝐴𝐷) ∨ (𝐵𝐶𝐵𝐷)) ↔ {𝐴, 𝐵} ≠ {𝐶, 𝐷}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  wne 2940  {cpr 4630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-v 3476  df-un 3953  df-sn 4629  df-pr 4631
This theorem is referenced by:  zlmodzxznm  47266
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