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Theorem prproe 4578
Description: For an element of a proper unordered pair of elements of a class 𝑉, there is another (different) element of the class 𝑉 which is an element of the proper pair. (Contributed by AV, 18-Dec-2021.)
Assertion
Ref Expression
prproe ((𝐶 ∈ {𝐴, 𝐵} ∧ 𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉)) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵})
Distinct variable groups:   𝑣,𝐴   𝑣,𝐵   𝑣,𝐶   𝑣,𝑉

Proof of Theorem prproe
StepHypRef Expression
1 elpri 4334 . . 3 (𝐶 ∈ {𝐴, 𝐵} → (𝐶 = 𝐴𝐶 = 𝐵))
2 simprrr 824 . . . . . . 7 ((𝐶 = 𝐴 ∧ (𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉))) → 𝐵𝑉)
3 necom 2977 . . . . . . . . . 10 (𝐴𝐵𝐵𝐴)
4 neeq2 2987 . . . . . . . . . . . 12 (𝐴 = 𝐶 → (𝐵𝐴𝐵𝐶))
54eqcoms 2760 . . . . . . . . . . 11 (𝐶 = 𝐴 → (𝐵𝐴𝐵𝐶))
65biimpcd 239 . . . . . . . . . 10 (𝐵𝐴 → (𝐶 = 𝐴𝐵𝐶))
73, 6sylbi 207 . . . . . . . . 9 (𝐴𝐵 → (𝐶 = 𝐴𝐵𝐶))
87adantr 472 . . . . . . . 8 ((𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉)) → (𝐶 = 𝐴𝐵𝐶))
98impcom 445 . . . . . . 7 ((𝐶 = 𝐴 ∧ (𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉))) → 𝐵𝐶)
10 eldifsn 4454 . . . . . . 7 (𝐵 ∈ (𝑉 ∖ {𝐶}) ↔ (𝐵𝑉𝐵𝐶))
112, 9, 10sylanbrc 701 . . . . . 6 ((𝐶 = 𝐴 ∧ (𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉))) → 𝐵 ∈ (𝑉 ∖ {𝐶}))
12 eleq1 2819 . . . . . . 7 (𝑣 = 𝐵 → (𝑣 ∈ {𝐴, 𝐵} ↔ 𝐵 ∈ {𝐴, 𝐵}))
1312adantl 473 . . . . . 6 (((𝐶 = 𝐴 ∧ (𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉))) ∧ 𝑣 = 𝐵) → (𝑣 ∈ {𝐴, 𝐵} ↔ 𝐵 ∈ {𝐴, 𝐵}))
14 prid2g 4432 . . . . . . . . 9 (𝐵𝑉𝐵 ∈ {𝐴, 𝐵})
1514adantl 473 . . . . . . . 8 ((𝐴𝑉𝐵𝑉) → 𝐵 ∈ {𝐴, 𝐵})
1615adantl 473 . . . . . . 7 ((𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉)) → 𝐵 ∈ {𝐴, 𝐵})
1716adantl 473 . . . . . 6 ((𝐶 = 𝐴 ∧ (𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉))) → 𝐵 ∈ {𝐴, 𝐵})
1811, 13, 17rspcedvd 3448 . . . . 5 ((𝐶 = 𝐴 ∧ (𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉))) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵})
1918ex 449 . . . 4 (𝐶 = 𝐴 → ((𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉)) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵}))
20 simprrl 823 . . . . . . 7 ((𝐶 = 𝐵 ∧ (𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉))) → 𝐴𝑉)
21 neeq2 2987 . . . . . . . . . . 11 (𝐵 = 𝐶 → (𝐴𝐵𝐴𝐶))
2221eqcoms 2760 . . . . . . . . . 10 (𝐶 = 𝐵 → (𝐴𝐵𝐴𝐶))
2322biimpcd 239 . . . . . . . . 9 (𝐴𝐵 → (𝐶 = 𝐵𝐴𝐶))
2423adantr 472 . . . . . . . 8 ((𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉)) → (𝐶 = 𝐵𝐴𝐶))
2524impcom 445 . . . . . . 7 ((𝐶 = 𝐵 ∧ (𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉))) → 𝐴𝐶)
26 eldifsn 4454 . . . . . . 7 (𝐴 ∈ (𝑉 ∖ {𝐶}) ↔ (𝐴𝑉𝐴𝐶))
2720, 25, 26sylanbrc 701 . . . . . 6 ((𝐶 = 𝐵 ∧ (𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉))) → 𝐴 ∈ (𝑉 ∖ {𝐶}))
28 eleq1 2819 . . . . . . 7 (𝑣 = 𝐴 → (𝑣 ∈ {𝐴, 𝐵} ↔ 𝐴 ∈ {𝐴, 𝐵}))
2928adantl 473 . . . . . 6 (((𝐶 = 𝐵 ∧ (𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉))) ∧ 𝑣 = 𝐴) → (𝑣 ∈ {𝐴, 𝐵} ↔ 𝐴 ∈ {𝐴, 𝐵}))
30 prid1g 4431 . . . . . . . . 9 (𝐴𝑉𝐴 ∈ {𝐴, 𝐵})
3130adantr 472 . . . . . . . 8 ((𝐴𝑉𝐵𝑉) → 𝐴 ∈ {𝐴, 𝐵})
3231adantl 473 . . . . . . 7 ((𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉)) → 𝐴 ∈ {𝐴, 𝐵})
3332adantl 473 . . . . . 6 ((𝐶 = 𝐵 ∧ (𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉))) → 𝐴 ∈ {𝐴, 𝐵})
3427, 29, 33rspcedvd 3448 . . . . 5 ((𝐶 = 𝐵 ∧ (𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉))) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵})
3534ex 449 . . . 4 (𝐶 = 𝐵 → ((𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉)) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵}))
3619, 35jaoi 393 . . 3 ((𝐶 = 𝐴𝐶 = 𝐵) → ((𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉)) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵}))
371, 36syl 17 . 2 (𝐶 ∈ {𝐴, 𝐵} → ((𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉)) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵}))
38373impib 1108 1 ((𝐶 ∈ {𝐴, 𝐵} ∧ 𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉)) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383  w3a 1072   = wceq 1624  wcel 2131  wne 2924  wrex 3043  cdif 3704  {csn 4313  {cpr 4315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-v 3334  df-dif 3710  df-un 3712  df-sn 4314  df-pr 4316
This theorem is referenced by:  edglnl  26229
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