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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 4854 . . 3 (((𝐴𝑈𝐵𝑉) ∧ (𝐶𝑋𝐷𝑌)) → (((𝐴𝐶𝐴𝐷) ∨ (𝐵𝐶𝐵𝐷)) → {𝐴, 𝐵} ≠ {𝐶, 𝐷}))
213adant3 1133 . 2 (((𝐴𝑈𝐵𝑉) ∧ (𝐶𝑋𝐷𝑌) ∧ 𝐴𝐵) → (((𝐴𝐶𝐴𝐷) ∨ (𝐵𝐶𝐵𝐷)) → {𝐴, 𝐵} ≠ {𝐶, 𝐷}))
3 ioran 986 . . . . 5 (¬ ((𝐴𝐶𝐴𝐷) ∨ (𝐵𝐶𝐵𝐷)) ↔ (¬ (𝐴𝐶𝐴𝐷) ∧ ¬ (𝐵𝐶𝐵𝐷)))
4 ianor 984 . . . . . . 7 (¬ (𝐴𝐶𝐴𝐷) ↔ (¬ 𝐴𝐶 ∨ ¬ 𝐴𝐷))
5 nne 2944 . . . . . . . 8 𝐴𝐶𝐴 = 𝐶)
6 nne 2944 . . . . . . . 8 𝐴𝐷𝐴 = 𝐷)
75, 6orbi12i 915 . . . . . . 7 ((¬ 𝐴𝐶 ∨ ¬ 𝐴𝐷) ↔ (𝐴 = 𝐶𝐴 = 𝐷))
84, 7bitri 275 . . . . . 6 (¬ (𝐴𝐶𝐴𝐷) ↔ (𝐴 = 𝐶𝐴 = 𝐷))
9 ianor 984 . . . . . . 7 (¬ (𝐵𝐶𝐵𝐷) ↔ (¬ 𝐵𝐶 ∨ ¬ 𝐵𝐷))
10 nne 2944 . . . . . . . 8 𝐵𝐶𝐵 = 𝐶)
11 nne 2944 . . . . . . . 8 𝐵𝐷𝐵 = 𝐷)
1210, 11orbi12i 915 . . . . . . 7 ((¬ 𝐵𝐶 ∨ ¬ 𝐵𝐷) ↔ (𝐵 = 𝐶𝐵 = 𝐷))
139, 12bitri 275 . . . . . 6 (¬ (𝐵𝐶𝐵𝐷) ↔ (𝐵 = 𝐶𝐵 = 𝐷))
148, 13anbi12i 628 . . . . 5 ((¬ (𝐴𝐶𝐴𝐷) ∧ ¬ (𝐵𝐶𝐵𝐷)) ↔ ((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐵 = 𝐶𝐵 = 𝐷)))
153, 14bitri 275 . . . 4 (¬ ((𝐴𝐶𝐴𝐷) ∨ (𝐵𝐶𝐵𝐷)) ↔ ((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐵 = 𝐶𝐵 = 𝐷)))
16 anddi 1013 . . . . 5 (((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐵 = 𝐶𝐵 = 𝐷)) ↔ (((𝐴 = 𝐶𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐷)) ∨ ((𝐴 = 𝐷𝐵 = 𝐶) ∨ (𝐴 = 𝐷𝐵 = 𝐷))))
17 eqtr3 2763 . . . . . . . . . 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 4735 . . . . . . . . . 10 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})
2120a1d 25 . . . . . . . . 9 ((𝐴 = 𝐶𝐵 = 𝐷) → (𝐴𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷}))
2219, 21jaoi 858 . . . . . . . 8 (((𝐴 = 𝐶𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐷)) → (𝐴𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷}))
23 preq12 4735 . . . . . . . . . . 11 ((𝐴 = 𝐷𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐷, 𝐶})
24 prcom 4732 . . . . . . . . . . 11 {𝐷, 𝐶} = {𝐶, 𝐷}
2523, 24eqtrdi 2793 . . . . . . . . . 10 ((𝐴 = 𝐷𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐶, 𝐷})
2625a1d 25 . . . . . . . . 9 ((𝐴 = 𝐷𝐵 = 𝐶) → (𝐴𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷}))
27 eqtr3 2763 . . . . . . . . . 10 ((𝐴 = 𝐷𝐵 = 𝐷) → 𝐴 = 𝐵)
2827, 18syl 17 . . . . . . . . 9 ((𝐴 = 𝐷𝐵 = 𝐷) → (𝐴𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷}))
2926, 28jaoi 858 . . . . . . . 8 (((𝐴 = 𝐷𝐵 = 𝐶) ∨ (𝐴 = 𝐷𝐵 = 𝐷)) → (𝐴𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷}))
3022, 29jaoi 858 . . . . . . 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 1136 . . . . 5 (((𝐴𝑈𝐵𝑉) ∧ (𝐶𝑋𝐷𝑌) ∧ 𝐴𝐵) → ((((𝐴 = 𝐶𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐷)) ∨ ((𝐴 = 𝐷𝐵 = 𝐶) ∨ (𝐴 = 𝐷𝐵 = 𝐷))) → {𝐴, 𝐵} = {𝐶, 𝐷}))
3316, 32biimtrid 242 . . . 4 (((𝐴𝑈𝐵𝑉) ∧ (𝐶𝑋𝐷𝑌) ∧ 𝐴𝐵) → (((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐵 = 𝐶𝐵 = 𝐷)) → {𝐴, 𝐵} = {𝐶, 𝐷}))
3415, 33biimtrid 242 . . 3 (((𝐴𝑈𝐵𝑉) ∧ (𝐶𝑋𝐷𝑌) ∧ 𝐴𝐵) → (¬ ((𝐴𝐶𝐴𝐷) ∨ (𝐵𝐶𝐵𝐷)) → {𝐴, 𝐵} = {𝐶, 𝐷}))
3534necon1ad 2957 . 2 (((𝐴𝑈𝐵𝑉) ∧ (𝐶𝑋𝐷𝑌) ∧ 𝐴𝐵) → ({𝐴, 𝐵} ≠ {𝐶, 𝐷} → ((𝐴𝐶𝐴𝐷) ∨ (𝐵𝐶𝐵𝐷))))
362, 35impbid 212 1 (((𝐴𝑈𝐵𝑉) ∧ (𝐶𝑋𝐷𝑌) ∧ 𝐴𝐵) → (((𝐴𝐶𝐴𝐷) ∨ (𝐵𝐶𝐵𝐷)) ↔ {𝐴, 𝐵} ≠ {𝐶, 𝐷}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  wne 2940  {cpr 4628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-v 3482  df-un 3956  df-sn 4627  df-pr 4629
This theorem is referenced by:  zlmodzxznm  48414
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