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Theorem prproe 4864
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 4609 . . 3 (𝐶 ∈ {𝐴, 𝐵} → (𝐶 = 𝐴𝐶 = 𝐵))
2 simprrr 781 . . . . . . 7 ((𝐶 = 𝐴 ∧ (𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉))) → 𝐵𝑉)
3 necom 2994 . . . . . . . . . 10 (𝐴𝐵𝐵𝐴)
4 neeq2 3004 . . . . . . . . . . . 12 (𝐴 = 𝐶 → (𝐵𝐴𝐵𝐶))
54eqcoms 2741 . . . . . . . . . . 11 (𝐶 = 𝐴 → (𝐵𝐴𝐵𝐶))
65biimpcd 249 . . . . . . . . . 10 (𝐵𝐴 → (𝐶 = 𝐴𝐵𝐶))
73, 6sylbi 216 . . . . . . . . 9 (𝐴𝐵 → (𝐶 = 𝐴𝐵𝐶))
87adantr 482 . . . . . . . 8 ((𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉)) → (𝐶 = 𝐴𝐵𝐶))
98impcom 409 . . . . . . 7 ((𝐶 = 𝐴 ∧ (𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉))) → 𝐵𝐶)
10 eldifsn 4748 . . . . . . 7 (𝐵 ∈ (𝑉 ∖ {𝐶}) ↔ (𝐵𝑉𝐵𝐶))
112, 9, 10sylanbrc 584 . . . . . 6 ((𝐶 = 𝐴 ∧ (𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉))) → 𝐵 ∈ (𝑉 ∖ {𝐶}))
12 eleq1 2822 . . . . . . 7 (𝑣 = 𝐵 → (𝑣 ∈ {𝐴, 𝐵} ↔ 𝐵 ∈ {𝐴, 𝐵}))
1312adantl 483 . . . . . 6 (((𝐶 = 𝐴 ∧ (𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉))) ∧ 𝑣 = 𝐵) → (𝑣 ∈ {𝐴, 𝐵} ↔ 𝐵 ∈ {𝐴, 𝐵}))
14 prid2g 4723 . . . . . . . . 9 (𝐵𝑉𝐵 ∈ {𝐴, 𝐵})
1514adantl 483 . . . . . . . 8 ((𝐴𝑉𝐵𝑉) → 𝐵 ∈ {𝐴, 𝐵})
1615adantl 483 . . . . . . 7 ((𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉)) → 𝐵 ∈ {𝐴, 𝐵})
1716adantl 483 . . . . . 6 ((𝐶 = 𝐴 ∧ (𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉))) → 𝐵 ∈ {𝐴, 𝐵})
1811, 13, 17rspcedvd 3582 . . . . 5 ((𝐶 = 𝐴 ∧ (𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉))) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵})
1918ex 414 . . . 4 (𝐶 = 𝐴 → ((𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉)) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵}))
20 simprrl 780 . . . . . . 7 ((𝐶 = 𝐵 ∧ (𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉))) → 𝐴𝑉)
21 neeq2 3004 . . . . . . . . . . 11 (𝐵 = 𝐶 → (𝐴𝐵𝐴𝐶))
2221eqcoms 2741 . . . . . . . . . 10 (𝐶 = 𝐵 → (𝐴𝐵𝐴𝐶))
2322biimpcd 249 . . . . . . . . 9 (𝐴𝐵 → (𝐶 = 𝐵𝐴𝐶))
2423adantr 482 . . . . . . . 8 ((𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉)) → (𝐶 = 𝐵𝐴𝐶))
2524impcom 409 . . . . . . 7 ((𝐶 = 𝐵 ∧ (𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉))) → 𝐴𝐶)
26 eldifsn 4748 . . . . . . 7 (𝐴 ∈ (𝑉 ∖ {𝐶}) ↔ (𝐴𝑉𝐴𝐶))
2720, 25, 26sylanbrc 584 . . . . . 6 ((𝐶 = 𝐵 ∧ (𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉))) → 𝐴 ∈ (𝑉 ∖ {𝐶}))
28 eleq1 2822 . . . . . . 7 (𝑣 = 𝐴 → (𝑣 ∈ {𝐴, 𝐵} ↔ 𝐴 ∈ {𝐴, 𝐵}))
2928adantl 483 . . . . . 6 (((𝐶 = 𝐵 ∧ (𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉))) ∧ 𝑣 = 𝐴) → (𝑣 ∈ {𝐴, 𝐵} ↔ 𝐴 ∈ {𝐴, 𝐵}))
30 prid1g 4722 . . . . . . . . 9 (𝐴𝑉𝐴 ∈ {𝐴, 𝐵})
3130adantr 482 . . . . . . . 8 ((𝐴𝑉𝐵𝑉) → 𝐴 ∈ {𝐴, 𝐵})
3231adantl 483 . . . . . . 7 ((𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉)) → 𝐴 ∈ {𝐴, 𝐵})
3332adantl 483 . . . . . 6 ((𝐶 = 𝐵 ∧ (𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉))) → 𝐴 ∈ {𝐴, 𝐵})
3427, 29, 33rspcedvd 3582 . . . . 5 ((𝐶 = 𝐵 ∧ (𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉))) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵})
3534ex 414 . . . 4 (𝐶 = 𝐵 → ((𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉)) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵}))
3619, 35jaoi 856 . . 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 1117 1 ((𝐶 ∈ {𝐴, 𝐵} ∧ 𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉)) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wo 846  w3a 1088   = wceq 1542  wcel 2107  wne 2940  wrex 3070  cdif 3908  {csn 4587  {cpr 4589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-v 3446  df-dif 3914  df-un 3916  df-sn 4588  df-pr 4590
This theorem is referenced by:  edglnl  28136
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