Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  prnebg Structured version   Visualization version   GIF version

Theorem prnebg 4357
 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 4356 . . 3 (((𝐴𝑈𝐵𝑉) ∧ (𝐶𝑋𝐷𝑌)) → (((𝐴𝐶𝐴𝐷) ∨ (𝐵𝐶𝐵𝐷)) → {𝐴, 𝐵} ≠ {𝐶, 𝐷}))
213adant3 1079 . 2 (((𝐴𝑈𝐵𝑉) ∧ (𝐶𝑋𝐷𝑌) ∧ 𝐴𝐵) → (((𝐴𝐶𝐴𝐷) ∨ (𝐵𝐶𝐵𝐷)) → {𝐴, 𝐵} ≠ {𝐶, 𝐷}))
3 ioran 511 . . . . 5 (¬ ((𝐴𝐶𝐴𝐷) ∨ (𝐵𝐶𝐵𝐷)) ↔ (¬ (𝐴𝐶𝐴𝐷) ∧ ¬ (𝐵𝐶𝐵𝐷)))
4 ianor 509 . . . . . . 7 (¬ (𝐴𝐶𝐴𝐷) ↔ (¬ 𝐴𝐶 ∨ ¬ 𝐴𝐷))
5 nne 2794 . . . . . . . 8 𝐴𝐶𝐴 = 𝐶)
6 nne 2794 . . . . . . . 8 𝐴𝐷𝐴 = 𝐷)
75, 6orbi12i 543 . . . . . . 7 ((¬ 𝐴𝐶 ∨ ¬ 𝐴𝐷) ↔ (𝐴 = 𝐶𝐴 = 𝐷))
84, 7bitri 264 . . . . . 6 (¬ (𝐴𝐶𝐴𝐷) ↔ (𝐴 = 𝐶𝐴 = 𝐷))
9 ianor 509 . . . . . . 7 (¬ (𝐵𝐶𝐵𝐷) ↔ (¬ 𝐵𝐶 ∨ ¬ 𝐵𝐷))
10 nne 2794 . . . . . . . 8 𝐵𝐶𝐵 = 𝐶)
11 nne 2794 . . . . . . . 8 𝐵𝐷𝐵 = 𝐷)
1210, 11orbi12i 543 . . . . . . 7 ((¬ 𝐵𝐶 ∨ ¬ 𝐵𝐷) ↔ (𝐵 = 𝐶𝐵 = 𝐷))
139, 12bitri 264 . . . . . 6 (¬ (𝐵𝐶𝐵𝐷) ↔ (𝐵 = 𝐶𝐵 = 𝐷))
148, 13anbi12i 732 . . . . 5 ((¬ (𝐴𝐶𝐴𝐷) ∧ ¬ (𝐵𝐶𝐵𝐷)) ↔ ((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐵 = 𝐶𝐵 = 𝐷)))
153, 14bitri 264 . . . 4 (¬ ((𝐴𝐶𝐴𝐷) ∨ (𝐵𝐶𝐵𝐷)) ↔ ((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐵 = 𝐶𝐵 = 𝐷)))
16 anddi 913 . . . . 5 (((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐵 = 𝐶𝐵 = 𝐷)) ↔ (((𝐴 = 𝐶𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐷)) ∨ ((𝐴 = 𝐷𝐵 = 𝐶) ∨ (𝐴 = 𝐷𝐵 = 𝐷))))
17 eqtr3 2642 . . . . . . . . . 10 ((𝐴 = 𝐶𝐵 = 𝐶) → 𝐴 = 𝐵)
18 eqneqall 2801 . . . . . . . . . 10 (𝐴 = 𝐵 → (𝐴𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷}))
1917, 18syl 17 . . . . . . . . 9 ((𝐴 = 𝐶𝐵 = 𝐶) → (𝐴𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷}))
20 preq12 4240 . . . . . . . . . 10 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})
2120a1d 25 . . . . . . . . 9 ((𝐴 = 𝐶𝐵 = 𝐷) → (𝐴𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷}))
2219, 21jaoi 394 . . . . . . . 8 (((𝐴 = 𝐶𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐷)) → (𝐴𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷}))
23 preq12 4240 . . . . . . . . . . 11 ((𝐴 = 𝐷𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐷, 𝐶})
24 prcom 4237 . . . . . . . . . . 11 {𝐷, 𝐶} = {𝐶, 𝐷}
2523, 24syl6eq 2671 . . . . . . . . . 10 ((𝐴 = 𝐷𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐶, 𝐷})
2625a1d 25 . . . . . . . . 9 ((𝐴 = 𝐷𝐵 = 𝐶) → (𝐴𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷}))
27 eqtr3 2642 . . . . . . . . . 10 ((𝐴 = 𝐷𝐵 = 𝐷) → 𝐴 = 𝐵)
2827, 18syl 17 . . . . . . . . 9 ((𝐴 = 𝐷𝐵 = 𝐷) → (𝐴𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷}))
2926, 28jaoi 394 . . . . . . . 8 (((𝐴 = 𝐷𝐵 = 𝐶) ∨ (𝐴 = 𝐷𝐵 = 𝐷)) → (𝐴𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷}))
3022, 29jaoi 394 . . . . . . 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 1082 . . . . 5 (((𝐴𝑈𝐵𝑉) ∧ (𝐶𝑋𝐷𝑌) ∧ 𝐴𝐵) → ((((𝐴 = 𝐶𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐷)) ∨ ((𝐴 = 𝐷𝐵 = 𝐶) ∨ (𝐴 = 𝐷𝐵 = 𝐷))) → {𝐴, 𝐵} = {𝐶, 𝐷}))
3316, 32syl5bi 232 . . . 4 (((𝐴𝑈𝐵𝑉) ∧ (𝐶𝑋𝐷𝑌) ∧ 𝐴𝐵) → (((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐵 = 𝐶𝐵 = 𝐷)) → {𝐴, 𝐵} = {𝐶, 𝐷}))
3415, 33syl5bi 232 . . 3 (((𝐴𝑈𝐵𝑉) ∧ (𝐶𝑋𝐷𝑌) ∧ 𝐴𝐵) → (¬ ((𝐴𝐶𝐴𝐷) ∨ (𝐵𝐶𝐵𝐷)) → {𝐴, 𝐵} = {𝐶, 𝐷}))
3534necon1ad 2807 . 2 (((𝐴𝑈𝐵𝑉) ∧ (𝐶𝑋𝐷𝑌) ∧ 𝐴𝐵) → ({𝐴, 𝐵} ≠ {𝐶, 𝐷} → ((𝐴𝐶𝐴𝐷) ∨ (𝐵𝐶𝐵𝐷))))
362, 35impbid 202 1 (((𝐴𝑈𝐵𝑉) ∧ (𝐶𝑋𝐷𝑌) ∧ 𝐴𝐵) → (((𝐴𝐶𝐴𝐷) ∨ (𝐵𝐶𝐵𝐷)) ↔ {𝐴, 𝐵} ≠ {𝐶, 𝐷}))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987   ≠ wne 2790  {cpr 4150 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-v 3188  df-un 3560  df-sn 4149  df-pr 4151 This theorem is referenced by:  zlmodzxznm  41574
 Copyright terms: Public domain W3C validator