Theorem prsprel 47969

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 47966	. . 3 ⊢ ({𝑋, 𝑌} ∈ (Pairs‘𝑉) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝑎, 𝑏})
2		preq12bg 4791	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑊) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ({𝑋, 𝑌} = {𝑎, 𝑏} ↔ ((𝑋 = 𝑎 ∧ 𝑌 = 𝑏) ∨ (𝑋 = 𝑏 ∧ 𝑌 = 𝑎))))
3		eleq1 2828	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑋 → (𝑎 ∈ 𝑉 ↔ 𝑋 ∈ 𝑉))
4	3	eqcoms 2748	. . . . . . . . . . . 12 ⊢ (𝑋 = 𝑎 → (𝑎 ∈ 𝑉 ↔ 𝑋 ∈ 𝑉))
5		eleq1 2828	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑌 → (𝑏 ∈ 𝑉 ↔ 𝑌 ∈ 𝑉))
6	5	eqcoms 2748	. . . . . . . . . . . 12 ⊢ (𝑌 = 𝑏 → (𝑏 ∈ 𝑉 ↔ 𝑌 ∈ 𝑉))
7	4, 6	bi2anan9 644	. . . . . . . . . . 11 ⊢ ((𝑋 = 𝑎 ∧ 𝑌 = 𝑏) → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ↔ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)))
8	7	biimpd 230	. . . . . . . . . 10 ⊢ ((𝑋 = 𝑎 ∧ 𝑌 = 𝑏) → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)))
9		eleq1 2828	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑋 → (𝑏 ∈ 𝑉 ↔ 𝑋 ∈ 𝑉))
10	9	eqcoms 2748	. . . . . . . . . . . . 13 ⊢ (𝑋 = 𝑏 → (𝑏 ∈ 𝑉 ↔ 𝑋 ∈ 𝑉))
11		eleq1 2828	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑌 → (𝑎 ∈ 𝑉 ↔ 𝑌 ∈ 𝑉))
12	11	eqcoms 2748	. . . . . . . . . . . . 13 ⊢ (𝑌 = 𝑎 → (𝑎 ∈ 𝑉 ↔ 𝑌 ∈ 𝑉))
13	10, 12	bi2anan9 644	. . . . . . . . . . . 12 ⊢ ((𝑋 = 𝑏 ∧ 𝑌 = 𝑎) → ((𝑏 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉) ↔ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)))
14	13	biimpd 230	. . . . . . . . . . 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 863	. . . . . . . . 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 241	. . . . . 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 3182	. . 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 207   ∧ wa 396   ∨ wo 853   = wceq 1547   ∈ wcel 2119  ∃wrex 3064  {cpr 4564  ‘cfv 6492  Pairscspr 47959

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6448  df-fun 6494  df-fv 6500  df-spr 47960

This theorem is referenced by:  prsssprel  47970  sprsymrelfolem2  47975