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Theorem prproe 4905
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 4649 . . 3 (𝐶 ∈ {𝐴, 𝐵} → (𝐶 = 𝐴𝐶 = 𝐵))
2 eleq1 2829 . . . . . 6 (𝑣 = 𝐵 → (𝑣 ∈ {𝐴, 𝐵} ↔ 𝐵 ∈ {𝐴, 𝐵}))
3 simprrr 782 . . . . . . 7 ((𝐶 = 𝐴 ∧ (𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉))) → 𝐵𝑉)
4 necom 2994 . . . . . . . . . 10 (𝐴𝐵𝐵𝐴)
5 neeq2 3004 . . . . . . . . . . . 12 (𝐴 = 𝐶 → (𝐵𝐴𝐵𝐶))
65eqcoms 2745 . . . . . . . . . . 11 (𝐶 = 𝐴 → (𝐵𝐴𝐵𝐶))
76biimpcd 249 . . . . . . . . . 10 (𝐵𝐴 → (𝐶 = 𝐴𝐵𝐶))
84, 7sylbi 217 . . . . . . . . 9 (𝐴𝐵 → (𝐶 = 𝐴𝐵𝐶))
98adantr 480 . . . . . . . 8 ((𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉)) → (𝐶 = 𝐴𝐵𝐶))
109impcom 407 . . . . . . 7 ((𝐶 = 𝐴 ∧ (𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉))) → 𝐵𝐶)
113, 10eldifsnd 4787 . . . . . 6 ((𝐶 = 𝐴 ∧ (𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉))) → 𝐵 ∈ (𝑉 ∖ {𝐶}))
12 prid2g 4761 . . . . . . . 8 (𝐵𝑉𝐵 ∈ {𝐴, 𝐵})
1312adantl 481 . . . . . . 7 ((𝐴𝑉𝐵𝑉) → 𝐵 ∈ {𝐴, 𝐵})
1413ad2antll 729 . . . . . 6 ((𝐶 = 𝐴 ∧ (𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉))) → 𝐵 ∈ {𝐴, 𝐵})
152, 11, 14rspcedvdw 3625 . . . . 5 ((𝐶 = 𝐴 ∧ (𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉))) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵})
1615ex 412 . . . 4 (𝐶 = 𝐴 → ((𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉)) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵}))
17 eleq1 2829 . . . . . 6 (𝑣 = 𝐴 → (𝑣 ∈ {𝐴, 𝐵} ↔ 𝐴 ∈ {𝐴, 𝐵}))
18 simprrl 781 . . . . . . 7 ((𝐶 = 𝐵 ∧ (𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉))) → 𝐴𝑉)
19 neeq2 3004 . . . . . . . . . . 11 (𝐵 = 𝐶 → (𝐴𝐵𝐴𝐶))
2019eqcoms 2745 . . . . . . . . . 10 (𝐶 = 𝐵 → (𝐴𝐵𝐴𝐶))
2120biimpcd 249 . . . . . . . . 9 (𝐴𝐵 → (𝐶 = 𝐵𝐴𝐶))
2221adantr 480 . . . . . . . 8 ((𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉)) → (𝐶 = 𝐵𝐴𝐶))
2322impcom 407 . . . . . . 7 ((𝐶 = 𝐵 ∧ (𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉))) → 𝐴𝐶)
2418, 23eldifsnd 4787 . . . . . 6 ((𝐶 = 𝐵 ∧ (𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉))) → 𝐴 ∈ (𝑉 ∖ {𝐶}))
25 prid1g 4760 . . . . . . . 8 (𝐴𝑉𝐴 ∈ {𝐴, 𝐵})
2625adantr 480 . . . . . . 7 ((𝐴𝑉𝐵𝑉) → 𝐴 ∈ {𝐴, 𝐵})
2726ad2antll 729 . . . . . 6 ((𝐶 = 𝐵 ∧ (𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉))) → 𝐴 ∈ {𝐴, 𝐵})
2817, 24, 27rspcedvdw 3625 . . . . 5 ((𝐶 = 𝐵 ∧ (𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉))) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵})
2928ex 412 . . . 4 (𝐶 = 𝐵 → ((𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉)) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵}))
3016, 29jaoi 858 . . 3 ((𝐶 = 𝐴𝐶 = 𝐵) → ((𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉)) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵}))
311, 30syl 17 . 2 (𝐶 ∈ {𝐴, 𝐵} → ((𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉)) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵}))
32313impib 1117 1 ((𝐶 ∈ {𝐴, 𝐵} ∧ 𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉)) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wrex 3070  cdif 3948  {csn 4626  {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-ral 3062  df-rex 3071  df-v 3482  df-dif 3954  df-un 3956  df-sn 4627  df-pr 4629
This theorem is referenced by:  edglnl  29160
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