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} ∧ 𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵})

Distinct variable groups:   𝑣,𝐴   𝑣,𝐵   𝑣,𝐶   𝑣,𝑉

Proof of Theorem prproe

Step	Hyp	Ref	Expression
1		elpri 4649	. . 3 ⊢ (𝐶 ∈ {𝐴, 𝐵} → (𝐶 = 𝐴 ∨ 𝐶 = 𝐵))
2		eleq1 2829	. . . . . 6 ⊢ (𝑣 = 𝐵 → (𝑣 ∈ {𝐴, 𝐵} ↔ 𝐵 ∈ {𝐴, 𝐵}))
3		simprrr 782	. . . . . . 7 ⊢ ((𝐶 = 𝐴 ∧ (𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))) → 𝐵 ∈ 𝑉)
4		necom 2994	. . . . . . . . . 10 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴)
5		neeq2 3004	. . . . . . . . . . . 12 ⊢ (𝐴 = 𝐶 → (𝐵 ≠ 𝐴 ↔ 𝐵 ≠ 𝐶))
6	5	eqcoms 2745	. . . . . . . . . . 11 ⊢ (𝐶 = 𝐴 → (𝐵 ≠ 𝐴 ↔ 𝐵 ≠ 𝐶))
7	6	biimpcd 249	. . . . . . . . . 10 ⊢ (𝐵 ≠ 𝐴 → (𝐶 = 𝐴 → 𝐵 ≠ 𝐶))
8	4, 7	sylbi 217	. . . . . . . . 9 ⊢ (𝐴 ≠ 𝐵 → (𝐶 = 𝐴 → 𝐵 ≠ 𝐶))
9	8	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐶 = 𝐴 → 𝐵 ≠ 𝐶))
10	9	impcom 407	. . . . . . 7 ⊢ ((𝐶 = 𝐴 ∧ (𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))) → 𝐵 ≠ 𝐶)
11	3, 10	eldifsnd 4787	. . . . . 6 ⊢ ((𝐶 = 𝐴 ∧ (𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))) → 𝐵 ∈ (𝑉 ∖ {𝐶}))
12		prid2g 4761	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ {𝐴, 𝐵})
13	12	adantl 481	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ {𝐴, 𝐵})
14	13	ad2antll 729	. . . . . 6 ⊢ ((𝐶 = 𝐴 ∧ (𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))) → 𝐵 ∈ {𝐴, 𝐵})
15	2, 11, 14	rspcedvdw 3625	. . . . 5 ⊢ ((𝐶 = 𝐴 ∧ (𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵})
16	15	ex 412	. . . 4 ⊢ (𝐶 = 𝐴 → ((𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵}))
17		eleq1 2829	. . . . . 6 ⊢ (𝑣 = 𝐴 → (𝑣 ∈ {𝐴, 𝐵} ↔ 𝐴 ∈ {𝐴, 𝐵}))
18		simprrl 781	. . . . . . 7 ⊢ ((𝐶 = 𝐵 ∧ (𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))) → 𝐴 ∈ 𝑉)
19		neeq2 3004	. . . . . . . . . . 11 ⊢ (𝐵 = 𝐶 → (𝐴 ≠ 𝐵 ↔ 𝐴 ≠ 𝐶))
20	19	eqcoms 2745	. . . . . . . . . 10 ⊢ (𝐶 = 𝐵 → (𝐴 ≠ 𝐵 ↔ 𝐴 ≠ 𝐶))
21	20	biimpcd 249	. . . . . . . . 9 ⊢ (𝐴 ≠ 𝐵 → (𝐶 = 𝐵 → 𝐴 ≠ 𝐶))
22	21	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐶 = 𝐵 → 𝐴 ≠ 𝐶))
23	22	impcom 407	. . . . . . 7 ⊢ ((𝐶 = 𝐵 ∧ (𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))) → 𝐴 ≠ 𝐶)
24	18, 23	eldifsnd 4787	. . . . . 6 ⊢ ((𝐶 = 𝐵 ∧ (𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))) → 𝐴 ∈ (𝑉 ∖ {𝐶}))
25		prid1g 4760	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵})
26	25	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ {𝐴, 𝐵})
27	26	ad2antll 729	. . . . . 6 ⊢ ((𝐶 = 𝐵 ∧ (𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))) → 𝐴 ∈ {𝐴, 𝐵})
28	17, 24, 27	rspcedvdw 3625	. . . . 5 ⊢ ((𝐶 = 𝐵 ∧ (𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵})
29	28	ex 412	. . . 4 ⊢ (𝐶 = 𝐵 → ((𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵}))
30	16, 29	jaoi 858	. . 3 ⊢ ((𝐶 = 𝐴 ∨ 𝐶 = 𝐵) → ((𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵}))
31	1, 30	syl 17	. 2 ⊢ (𝐶 ∈ {𝐴, 𝐵} → ((𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵}))
32	31	3impib 1117	1 ⊢ ((𝐶 ∈ {𝐴, 𝐵} ∧ 𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵})

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