Theorem prsprel 47474

Description: The elements of a pair from the set of all unordered pairs over a given set 𝑉 are elements of the set 𝑉. (Contributed by AV, 22-Nov-2021.)

Assertion

Ref	Expression
prsprel	⊢ (({𝑋, 𝑌} ∈ (Pairs‘𝑉) ∧ (𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑊)) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉))

Proof of Theorem prsprel

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sprel 47471	. . 3 ⊢ ({𝑋, 𝑌} ∈ (Pairs‘𝑉) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝑎, 𝑏})
2		preq12bg 4853	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑊) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ({𝑋, 𝑌} = {𝑎, 𝑏} ↔ ((𝑋 = 𝑎 ∧ 𝑌 = 𝑏) ∨ (𝑋 = 𝑏 ∧ 𝑌 = 𝑎))))
3		eleq1 2829	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑋 → (𝑎 ∈ 𝑉 ↔ 𝑋 ∈ 𝑉))
4	3	eqcoms 2745	. . . . . . . . . . . 12 ⊢ (𝑋 = 𝑎 → (𝑎 ∈ 𝑉 ↔ 𝑋 ∈ 𝑉))
5		eleq1 2829	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑌 → (𝑏 ∈ 𝑉 ↔ 𝑌 ∈ 𝑉))
6	5	eqcoms 2745	. . . . . . . . . . . 12 ⊢ (𝑌 = 𝑏 → (𝑏 ∈ 𝑉 ↔ 𝑌 ∈ 𝑉))
7	4, 6	bi2anan9 638	. . . . . . . . . . 11 ⊢ ((𝑋 = 𝑎 ∧ 𝑌 = 𝑏) → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ↔ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)))
8	7	biimpd 229	. . . . . . . . . 10 ⊢ ((𝑋 = 𝑎 ∧ 𝑌 = 𝑏) → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)))
9		eleq1 2829	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑋 → (𝑏 ∈ 𝑉 ↔ 𝑋 ∈ 𝑉))
10	9	eqcoms 2745	. . . . . . . . . . . . 13 ⊢ (𝑋 = 𝑏 → (𝑏 ∈ 𝑉 ↔ 𝑋 ∈ 𝑉))
11		eleq1 2829	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑌 → (𝑎 ∈ 𝑉 ↔ 𝑌 ∈ 𝑉))
12	11	eqcoms 2745	. . . . . . . . . . . . 13 ⊢ (𝑌 = 𝑎 → (𝑎 ∈ 𝑉 ↔ 𝑌 ∈ 𝑉))
13	10, 12	bi2anan9 638	. . . . . . . . . . . 12 ⊢ ((𝑋 = 𝑏 ∧ 𝑌 = 𝑎) → ((𝑏 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉) ↔ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)))
14	13	biimpd 229	. . . . . . . . . . 11 ⊢ ((𝑋 = 𝑏 ∧ 𝑌 = 𝑎) → ((𝑏 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)))
15	14	ancomsd 465	. . . . . . . . . 10 ⊢ ((𝑋 = 𝑏 ∧ 𝑌 = 𝑎) → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)))
16	8, 15	jaoi 858	. . . . . . . . 9 ⊢ (((𝑋 = 𝑎 ∧ 𝑌 = 𝑏) ∨ (𝑋 = 𝑏 ∧ 𝑌 = 𝑎)) → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)))
17	16	com12 32	. . . . . . . 8 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (((𝑋 = 𝑎 ∧ 𝑌 = 𝑏) ∨ (𝑋 = 𝑏 ∧ 𝑌 = 𝑎)) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)))
18	17	adantl 481	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑊) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (((𝑋 = 𝑎 ∧ 𝑌 = 𝑏) ∨ (𝑋 = 𝑏 ∧ 𝑌 = 𝑎)) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)))
19	2, 18	sylbid 240	. . . . . 6 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑊) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ({𝑋, 𝑌} = {𝑎, 𝑏} → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)))
20	19	expcom 413	. . . . 5 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑊) → ({𝑋, 𝑌} = {𝑎, 𝑏} → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉))))
21	20	com23 86	. . . 4 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ({𝑋, 𝑌} = {𝑎, 𝑏} → ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑊) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉))))
22	21	rexlimivv 3201	. . 3 ⊢ (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝑎, 𝑏} → ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑊) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)))
23	1, 22	syl 17	. 2 ⊢ ({𝑋, 𝑌} ∈ (Pairs‘𝑉) → ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑊) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)))
24	23	imp 406	1 ⊢ (({𝑋, 𝑌} ∈ (Pairs‘𝑉) ∧ (𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑊)) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1540   ∈ wcel 2108  ∃wrex 3070  {cpr 4628  ‘cfv 6561  Pairscspr 47464

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-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-spr 47465

This theorem is referenced by:  prsssprel  47475  sprsymrelfolem2  47480