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Theorem pr1eqbg 4779
Description: A (proper) pair is equal to another (maybe improper) pair containing one element of the first pair if and only if the other element of the first pair is contained in the second pair. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
Assertion
Ref Expression
pr1eqbg (((𝐴𝑈𝐵𝑉𝐶𝑋) ∧ 𝐴𝐵) → (𝐴 = 𝐶 ↔ {𝐴, 𝐵} = {𝐵, 𝐶}))

Proof of Theorem pr1eqbg
StepHypRef Expression
1 eqid 2819 . . . . 5 𝐵 = 𝐵
21biantru 532 . . . 4 (𝐴 = 𝐶 ↔ (𝐴 = 𝐶𝐵 = 𝐵))
32orbi2i 909 . . 3 (((𝐴 = 𝐵𝐵 = 𝐶) ∨ 𝐴 = 𝐶) ↔ ((𝐴 = 𝐵𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐵)))
43a1i 11 . 2 (((𝐴𝑈𝐵𝑉𝐶𝑋) ∧ 𝐴𝐵) → (((𝐴 = 𝐵𝐵 = 𝐶) ∨ 𝐴 = 𝐶) ↔ ((𝐴 = 𝐵𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐵))))
5 neneq 3020 . . . . 5 (𝐴𝐵 → ¬ 𝐴 = 𝐵)
65adantl 484 . . . 4 (((𝐴𝑈𝐵𝑉𝐶𝑋) ∧ 𝐴𝐵) → ¬ 𝐴 = 𝐵)
76intnanrd 492 . . 3 (((𝐴𝑈𝐵𝑉𝐶𝑋) ∧ 𝐴𝐵) → ¬ (𝐴 = 𝐵𝐵 = 𝐶))
8 biorf 933 . . 3 (¬ (𝐴 = 𝐵𝐵 = 𝐶) → (𝐴 = 𝐶 ↔ ((𝐴 = 𝐵𝐵 = 𝐶) ∨ 𝐴 = 𝐶)))
97, 8syl 17 . 2 (((𝐴𝑈𝐵𝑉𝐶𝑋) ∧ 𝐴𝐵) → (𝐴 = 𝐶 ↔ ((𝐴 = 𝐵𝐵 = 𝐶) ∨ 𝐴 = 𝐶)))
10 3simpa 1143 . . . . 5 ((𝐴𝑈𝐵𝑉𝐶𝑋) → (𝐴𝑈𝐵𝑉))
11 3simpc 1145 . . . . 5 ((𝐴𝑈𝐵𝑉𝐶𝑋) → (𝐵𝑉𝐶𝑋))
1210, 11jca 514 . . . 4 ((𝐴𝑈𝐵𝑉𝐶𝑋) → ((𝐴𝑈𝐵𝑉) ∧ (𝐵𝑉𝐶𝑋)))
1312adantr 483 . . 3 (((𝐴𝑈𝐵𝑉𝐶𝑋) ∧ 𝐴𝐵) → ((𝐴𝑈𝐵𝑉) ∧ (𝐵𝑉𝐶𝑋)))
14 preq12bg 4776 . . 3 (((𝐴𝑈𝐵𝑉) ∧ (𝐵𝑉𝐶𝑋)) → ({𝐴, 𝐵} = {𝐵, 𝐶} ↔ ((𝐴 = 𝐵𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐵))))
1513, 14syl 17 . 2 (((𝐴𝑈𝐵𝑉𝐶𝑋) ∧ 𝐴𝐵) → ({𝐴, 𝐵} = {𝐵, 𝐶} ↔ ((𝐴 = 𝐵𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐵))))
164, 9, 153bitr4d 313 1 (((𝐴𝑈𝐵𝑉𝐶𝑋) ∧ 𝐴𝐵) → (𝐴 = 𝐶 ↔ {𝐴, 𝐵} = {𝐵, 𝐶}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  wo 843  w3a 1082   = wceq 1531  wcel 2108  wne 3014  {cpr 4561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-v 3495  df-un 3939  df-sn 4560  df-pr 4562
This theorem is referenced by:  pr1nebg  4780
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