Theorem prproe 4838

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 4608	. . 3 ⊢ (𝐶 ∈ {𝐴, 𝐵} → (𝐶 = 𝐴 ∨ 𝐶 = 𝐵))
2		eleq1 2852	. . . . . 6 ⊢ (𝑣 = 𝐵 → (𝑣 ∈ {𝐴, 𝐵} ↔ 𝐵 ∈ {𝐴, 𝐵}))
3		simprrr 794	. . . . . . 7 ⊢ ((𝐶 = 𝐴 ∧ (𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))) → 𝐵 ∈ 𝑉)
4		necom 3012	. . . . . . . . . 10 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴)
5		neeq2 3022	. . . . . . . . . . . 12 ⊢ (𝐴 = 𝐶 → (𝐵 ≠ 𝐴 ↔ 𝐵 ≠ 𝐶))
6	5	eqcoms 2771	. . . . . . . . . . 11 ⊢ (𝐶 = 𝐴 → (𝐵 ≠ 𝐴 ↔ 𝐵 ≠ 𝐶))
7	6	biimpcd 252	. . . . . . . . . 10 ⊢ (𝐵 ≠ 𝐴 → (𝐶 = 𝐴 → 𝐵 ≠ 𝐶))
8	4, 7	sylbi 220	. . . . . . . . 9 ⊢ (𝐴 ≠ 𝐵 → (𝐶 = 𝐴 → 𝐵 ≠ 𝐶))
9	8	adantr 486	. . . . . . . 8 ⊢ ((𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐶 = 𝐴 → 𝐵 ≠ 𝐶))
10	9	impcom 413	. . . . . . 7 ⊢ ((𝐶 = 𝐴 ∧ (𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))) → 𝐵 ≠ 𝐶)
11	3, 10	eldifsnd 4752	. . . . . 6 ⊢ ((𝐶 = 𝐴 ∧ (𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))) → 𝐵 ∈ (𝑉 ∖ {𝐶}))
12		prid2g 4724	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ {𝐴, 𝐵})
13	12	adantl 487	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ {𝐴, 𝐵})
14	13	ad2antll 742	. . . . . 6 ⊢ ((𝐶 = 𝐴 ∧ (𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))) → 𝐵 ∈ {𝐴, 𝐵})
15	2, 11, 14	rspcedvdw 3583	. . . . 5 ⊢ ((𝐶 = 𝐴 ∧ (𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵})
16	15	ex 418	. . . 4 ⊢ (𝐶 = 𝐴 → ((𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵}))
17		eleq1 2852	. . . . . 6 ⊢ (𝑣 = 𝐴 → (𝑣 ∈ {𝐴, 𝐵} ↔ 𝐴 ∈ {𝐴, 𝐵}))
18		simprrl 793	. . . . . . 7 ⊢ ((𝐶 = 𝐵 ∧ (𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))) → 𝐴 ∈ 𝑉)
19		neeq2 3022	. . . . . . . . . . 11 ⊢ (𝐵 = 𝐶 → (𝐴 ≠ 𝐵 ↔ 𝐴 ≠ 𝐶))
20	19	eqcoms 2771	. . . . . . . . . 10 ⊢ (𝐶 = 𝐵 → (𝐴 ≠ 𝐵 ↔ 𝐴 ≠ 𝐶))
21	20	biimpcd 252	. . . . . . . . 9 ⊢ (𝐴 ≠ 𝐵 → (𝐶 = 𝐵 → 𝐴 ≠ 𝐶))
22	21	adantr 486	. . . . . . . 8 ⊢ ((𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐶 = 𝐵 → 𝐴 ≠ 𝐶))
23	22	impcom 413	. . . . . . 7 ⊢ ((𝐶 = 𝐵 ∧ (𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))) → 𝐴 ≠ 𝐶)
24	18, 23	eldifsnd 4752	. . . . . 6 ⊢ ((𝐶 = 𝐵 ∧ (𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))) → 𝐴 ∈ (𝑉 ∖ {𝐶}))
25		prid1g 4723	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵})
26	25	adantr 486	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ {𝐴, 𝐵})
27	26	ad2antll 742	. . . . . 6 ⊢ ((𝐶 = 𝐵 ∧ (𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))) → 𝐴 ∈ {𝐴, 𝐵})
28	17, 24, 27	rspcedvdw 3583	. . . . 5 ⊢ ((𝐶 = 𝐵 ∧ (𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵})
29	28	ex 418	. . . 4 ⊢ (𝐶 = 𝐵 → ((𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵}))
30	16, 29	jaoi 871	. . 3 ⊢ ((𝐶 = 𝐴 ∨ 𝐶 = 𝐵) → ((𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵}))
31	1, 30	syl 18	. 2 ⊢ (𝐶 ∈ {𝐴, 𝐵} → ((𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵}))
32	31	3impib 1134	1 ⊢ ((𝐶 ∈ {𝐴, 𝐵} ∧ 𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∃wrex 3090   ∖ cdif 4048  {csn 4588  {cpr 4590

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-v 3456  df-dif 4050  df-un 4154  df-sn 4589  df-pr 4591

This theorem is used by:  edglnl  29778