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Theorem 3elpr2eq 4587
Description: If there are three elements in a proper unordered pair, and two of them are different from the third one, the two must be equal. (Contributed by AV, 19-Dec-2021.)
Assertion
Ref Expression
3elpr2eq (((𝑋 ∈ {𝐴, 𝐵} ∧ 𝑌 ∈ {𝐴, 𝐵} ∧ 𝑍 ∈ {𝐴, 𝐵}) ∧ (𝑌𝑋𝑍𝑋)) → 𝑌 = 𝑍)

Proof of Theorem 3elpr2eq
StepHypRef Expression
1 elpri 4342 . . 3 (𝑋 ∈ {𝐴, 𝐵} → (𝑋 = 𝐴𝑋 = 𝐵))
2 elpri 4342 . . 3 (𝑌 ∈ {𝐴, 𝐵} → (𝑌 = 𝐴𝑌 = 𝐵))
3 elpri 4342 . . 3 (𝑍 ∈ {𝐴, 𝐵} → (𝑍 = 𝐴𝑍 = 𝐵))
4 eqtr3 2781 . . . . . . . . . . 11 ((𝑍 = 𝐴𝑋 = 𝐴) → 𝑍 = 𝑋)
5 eqneqall 2943 . . . . . . . . . . 11 (𝑍 = 𝑋 → (𝑍𝑋𝑌 = 𝑍))
64, 5syl 17 . . . . . . . . . 10 ((𝑍 = 𝐴𝑋 = 𝐴) → (𝑍𝑋𝑌 = 𝑍))
76adantld 484 . . . . . . . . 9 ((𝑍 = 𝐴𝑋 = 𝐴) → ((𝑌𝑋𝑍𝑋) → 𝑌 = 𝑍))
87ex 449 . . . . . . . 8 (𝑍 = 𝐴 → (𝑋 = 𝐴 → ((𝑌𝑋𝑍𝑋) → 𝑌 = 𝑍)))
98a1d 25 . . . . . . 7 (𝑍 = 𝐴 → ((𝑌 = 𝐴𝑌 = 𝐵) → (𝑋 = 𝐴 → ((𝑌𝑋𝑍𝑋) → 𝑌 = 𝑍))))
10 eqtr3 2781 . . . . . . . . . . . . 13 ((𝑌 = 𝐴𝑋 = 𝐴) → 𝑌 = 𝑋)
11 eqneqall 2943 . . . . . . . . . . . . 13 (𝑌 = 𝑋 → (𝑌𝑋 → (𝑍𝑋𝑌 = 𝑍)))
1210, 11syl 17 . . . . . . . . . . . 12 ((𝑌 = 𝐴𝑋 = 𝐴) → (𝑌𝑋 → (𝑍𝑋𝑌 = 𝑍)))
1312impd 446 . . . . . . . . . . 11 ((𝑌 = 𝐴𝑋 = 𝐴) → ((𝑌𝑋𝑍𝑋) → 𝑌 = 𝑍))
1413ex 449 . . . . . . . . . 10 (𝑌 = 𝐴 → (𝑋 = 𝐴 → ((𝑌𝑋𝑍𝑋) → 𝑌 = 𝑍)))
1514a1d 25 . . . . . . . . 9 (𝑌 = 𝐴 → (𝑍 = 𝐵 → (𝑋 = 𝐴 → ((𝑌𝑋𝑍𝑋) → 𝑌 = 𝑍))))
16 eqtr3 2781 . . . . . . . . . . 11 ((𝑌 = 𝐵𝑍 = 𝐵) → 𝑌 = 𝑍)
17162a1d 26 . . . . . . . . . 10 ((𝑌 = 𝐵𝑍 = 𝐵) → (𝑋 = 𝐴 → ((𝑌𝑋𝑍𝑋) → 𝑌 = 𝑍)))
1817ex 449 . . . . . . . . 9 (𝑌 = 𝐵 → (𝑍 = 𝐵 → (𝑋 = 𝐴 → ((𝑌𝑋𝑍𝑋) → 𝑌 = 𝑍))))
1915, 18jaoi 393 . . . . . . . 8 ((𝑌 = 𝐴𝑌 = 𝐵) → (𝑍 = 𝐵 → (𝑋 = 𝐴 → ((𝑌𝑋𝑍𝑋) → 𝑌 = 𝑍))))
2019com12 32 . . . . . . 7 (𝑍 = 𝐵 → ((𝑌 = 𝐴𝑌 = 𝐵) → (𝑋 = 𝐴 → ((𝑌𝑋𝑍𝑋) → 𝑌 = 𝑍))))
219, 20jaoi 393 . . . . . 6 ((𝑍 = 𝐴𝑍 = 𝐵) → ((𝑌 = 𝐴𝑌 = 𝐵) → (𝑋 = 𝐴 → ((𝑌𝑋𝑍𝑋) → 𝑌 = 𝑍))))
2221com13 88 . . . . 5 (𝑋 = 𝐴 → ((𝑌 = 𝐴𝑌 = 𝐵) → ((𝑍 = 𝐴𝑍 = 𝐵) → ((𝑌𝑋𝑍𝑋) → 𝑌 = 𝑍))))
23 eqtr3 2781 . . . . . . . . . . 11 ((𝑌 = 𝐴𝑍 = 𝐴) → 𝑌 = 𝑍)
24232a1d 26 . . . . . . . . . 10 ((𝑌 = 𝐴𝑍 = 𝐴) → (𝑋 = 𝐵 → ((𝑌𝑋𝑍𝑋) → 𝑌 = 𝑍)))
2524ex 449 . . . . . . . . 9 (𝑌 = 𝐴 → (𝑍 = 𝐴 → (𝑋 = 𝐵 → ((𝑌𝑋𝑍𝑋) → 𝑌 = 𝑍))))
26 eqtr3 2781 . . . . . . . . . . . . 13 ((𝑌 = 𝐵𝑋 = 𝐵) → 𝑌 = 𝑋)
2726, 11syl 17 . . . . . . . . . . . 12 ((𝑌 = 𝐵𝑋 = 𝐵) → (𝑌𝑋 → (𝑍𝑋𝑌 = 𝑍)))
2827impd 446 . . . . . . . . . . 11 ((𝑌 = 𝐵𝑋 = 𝐵) → ((𝑌𝑋𝑍𝑋) → 𝑌 = 𝑍))
2928ex 449 . . . . . . . . . 10 (𝑌 = 𝐵 → (𝑋 = 𝐵 → ((𝑌𝑋𝑍𝑋) → 𝑌 = 𝑍)))
3029a1d 25 . . . . . . . . 9 (𝑌 = 𝐵 → (𝑍 = 𝐴 → (𝑋 = 𝐵 → ((𝑌𝑋𝑍𝑋) → 𝑌 = 𝑍))))
3125, 30jaoi 393 . . . . . . . 8 ((𝑌 = 𝐴𝑌 = 𝐵) → (𝑍 = 𝐴 → (𝑋 = 𝐵 → ((𝑌𝑋𝑍𝑋) → 𝑌 = 𝑍))))
3231com12 32 . . . . . . 7 (𝑍 = 𝐴 → ((𝑌 = 𝐴𝑌 = 𝐵) → (𝑋 = 𝐵 → ((𝑌𝑋𝑍𝑋) → 𝑌 = 𝑍))))
33 eqtr3 2781 . . . . . . . . . . 11 ((𝑍 = 𝐵𝑋 = 𝐵) → 𝑍 = 𝑋)
3433, 5syl 17 . . . . . . . . . 10 ((𝑍 = 𝐵𝑋 = 𝐵) → (𝑍𝑋𝑌 = 𝑍))
3534adantld 484 . . . . . . . . 9 ((𝑍 = 𝐵𝑋 = 𝐵) → ((𝑌𝑋𝑍𝑋) → 𝑌 = 𝑍))
3635ex 449 . . . . . . . 8 (𝑍 = 𝐵 → (𝑋 = 𝐵 → ((𝑌𝑋𝑍𝑋) → 𝑌 = 𝑍)))
3736a1d 25 . . . . . . 7 (𝑍 = 𝐵 → ((𝑌 = 𝐴𝑌 = 𝐵) → (𝑋 = 𝐵 → ((𝑌𝑋𝑍𝑋) → 𝑌 = 𝑍))))
3832, 37jaoi 393 . . . . . 6 ((𝑍 = 𝐴𝑍 = 𝐵) → ((𝑌 = 𝐴𝑌 = 𝐵) → (𝑋 = 𝐵 → ((𝑌𝑋𝑍𝑋) → 𝑌 = 𝑍))))
3938com13 88 . . . . 5 (𝑋 = 𝐵 → ((𝑌 = 𝐴𝑌 = 𝐵) → ((𝑍 = 𝐴𝑍 = 𝐵) → ((𝑌𝑋𝑍𝑋) → 𝑌 = 𝑍))))
4022, 39jaoi 393 . . . 4 ((𝑋 = 𝐴𝑋 = 𝐵) → ((𝑌 = 𝐴𝑌 = 𝐵) → ((𝑍 = 𝐴𝑍 = 𝐵) → ((𝑌𝑋𝑍𝑋) → 𝑌 = 𝑍))))
41403imp 1102 . . 3 (((𝑋 = 𝐴𝑋 = 𝐵) ∧ (𝑌 = 𝐴𝑌 = 𝐵) ∧ (𝑍 = 𝐴𝑍 = 𝐵)) → ((𝑌𝑋𝑍𝑋) → 𝑌 = 𝑍))
421, 2, 3, 41syl3an 1164 . 2 ((𝑋 ∈ {𝐴, 𝐵} ∧ 𝑌 ∈ {𝐴, 𝐵} ∧ 𝑍 ∈ {𝐴, 𝐵}) → ((𝑌𝑋𝑍𝑋) → 𝑌 = 𝑍))
4342imp 444 1 (((𝑋 ∈ {𝐴, 𝐵} ∧ 𝑌 ∈ {𝐴, 𝐵} ∧ 𝑍 ∈ {𝐴, 𝐵}) ∧ (𝑌𝑋𝑍𝑋)) → 𝑌 = 𝑍)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382  wa 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932  {cpr 4323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-v 3342  df-un 3720  df-sn 4322  df-pr 4324
This theorem is referenced by:  numedglnl  26238
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