Theorem prsprel 44939

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 44936	. . 3 ⊢ ({𝑋, 𝑌} ∈ (Pairs‘𝑉) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝑎, 𝑏})
2		preq12bg 4784	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑊) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ({𝑋, 𝑌} = {𝑎, 𝑏} ↔ ((𝑋 = 𝑎 ∧ 𝑌 = 𝑏) ∨ (𝑋 = 𝑏 ∧ 𝑌 = 𝑎))))
3		eleq1 2826	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑋 → (𝑎 ∈ 𝑉 ↔ 𝑋 ∈ 𝑉))
4	3	eqcoms 2746	. . . . . . . . . . . 12 ⊢ (𝑋 = 𝑎 → (𝑎 ∈ 𝑉 ↔ 𝑋 ∈ 𝑉))
5		eleq1 2826	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑌 → (𝑏 ∈ 𝑉 ↔ 𝑌 ∈ 𝑉))
6	5	eqcoms 2746	. . . . . . . . . . . 12 ⊢ (𝑌 = 𝑏 → (𝑏 ∈ 𝑉 ↔ 𝑌 ∈ 𝑉))
7	4, 6	bi2anan9 636	. . . . . . . . . . 11 ⊢ ((𝑋 = 𝑎 ∧ 𝑌 = 𝑏) → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ↔ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)))
8	7	biimpd 228	. . . . . . . . . 10 ⊢ ((𝑋 = 𝑎 ∧ 𝑌 = 𝑏) → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)))
9		eleq1 2826	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑋 → (𝑏 ∈ 𝑉 ↔ 𝑋 ∈ 𝑉))
10	9	eqcoms 2746	. . . . . . . . . . . . 13 ⊢ (𝑋 = 𝑏 → (𝑏 ∈ 𝑉 ↔ 𝑋 ∈ 𝑉))
11		eleq1 2826	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑌 → (𝑎 ∈ 𝑉 ↔ 𝑌 ∈ 𝑉))
12	11	eqcoms 2746	. . . . . . . . . . . . 13 ⊢ (𝑌 = 𝑎 → (𝑎 ∈ 𝑉 ↔ 𝑌 ∈ 𝑉))
13	10, 12	bi2anan9 636	. . . . . . . . . . . 12 ⊢ ((𝑋 = 𝑏 ∧ 𝑌 = 𝑎) → ((𝑏 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉) ↔ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)))
14	13	biimpd 228	. . . . . . . . . . 11 ⊢ ((𝑋 = 𝑏 ∧ 𝑌 = 𝑎) → ((𝑏 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)))
15	14	ancomsd 466	. . . . . . . . . 10 ⊢ ((𝑋 = 𝑏 ∧ 𝑌 = 𝑎) → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)))
16	8, 15	jaoi 854	. . . . . . . . 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 482	. . . . . . 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 239	. . . . . 6 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑊) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ({𝑋, 𝑌} = {𝑎, 𝑏} → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)))
20	19	expcom 414	. . . . 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 3221	. . 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 407	1 ⊢ (({𝑋, 𝑌} ∈ (Pairs‘𝑉) ∧ (𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑊)) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   = wceq 1539   ∈ wcel 2106  ∃wrex 3065  {cpr 4563  ‘cfv 6433  Pairscspr 44929

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-spr 44930

This theorem is referenced by:  prsssprel  44940  sprsymrelfolem2  44945