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Theorem prsprel 44047
 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) ∧ (𝑋𝑈𝑌𝑊)) → (𝑋𝑉𝑌𝑉))

Proof of Theorem prsprel
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sprel 44044 . . 3 ({𝑋, 𝑌} ∈ (Pairs‘𝑉) → ∃𝑎𝑉𝑏𝑉 {𝑋, 𝑌} = {𝑎, 𝑏})
2 preq12bg 4744 . . . . . . 7 (((𝑋𝑈𝑌𝑊) ∧ (𝑎𝑉𝑏𝑉)) → ({𝑋, 𝑌} = {𝑎, 𝑏} ↔ ((𝑋 = 𝑎𝑌 = 𝑏) ∨ (𝑋 = 𝑏𝑌 = 𝑎))))
3 eleq1 2877 . . . . . . . . . . . . 13 (𝑎 = 𝑋 → (𝑎𝑉𝑋𝑉))
43eqcoms 2806 . . . . . . . . . . . 12 (𝑋 = 𝑎 → (𝑎𝑉𝑋𝑉))
5 eleq1 2877 . . . . . . . . . . . . 13 (𝑏 = 𝑌 → (𝑏𝑉𝑌𝑉))
65eqcoms 2806 . . . . . . . . . . . 12 (𝑌 = 𝑏 → (𝑏𝑉𝑌𝑉))
74, 6bi2anan9 638 . . . . . . . . . . 11 ((𝑋 = 𝑎𝑌 = 𝑏) → ((𝑎𝑉𝑏𝑉) ↔ (𝑋𝑉𝑌𝑉)))
87biimpd 232 . . . . . . . . . 10 ((𝑋 = 𝑎𝑌 = 𝑏) → ((𝑎𝑉𝑏𝑉) → (𝑋𝑉𝑌𝑉)))
9 eleq1 2877 . . . . . . . . . . . . . 14 (𝑏 = 𝑋 → (𝑏𝑉𝑋𝑉))
109eqcoms 2806 . . . . . . . . . . . . 13 (𝑋 = 𝑏 → (𝑏𝑉𝑋𝑉))
11 eleq1 2877 . . . . . . . . . . . . . 14 (𝑎 = 𝑌 → (𝑎𝑉𝑌𝑉))
1211eqcoms 2806 . . . . . . . . . . . . 13 (𝑌 = 𝑎 → (𝑎𝑉𝑌𝑉))
1310, 12bi2anan9 638 . . . . . . . . . . . 12 ((𝑋 = 𝑏𝑌 = 𝑎) → ((𝑏𝑉𝑎𝑉) ↔ (𝑋𝑉𝑌𝑉)))
1413biimpd 232 . . . . . . . . . . 11 ((𝑋 = 𝑏𝑌 = 𝑎) → ((𝑏𝑉𝑎𝑉) → (𝑋𝑉𝑌𝑉)))
1514ancomsd 469 . . . . . . . . . 10 ((𝑋 = 𝑏𝑌 = 𝑎) → ((𝑎𝑉𝑏𝑉) → (𝑋𝑉𝑌𝑉)))
168, 15jaoi 854 . . . . . . . . 9 (((𝑋 = 𝑎𝑌 = 𝑏) ∨ (𝑋 = 𝑏𝑌 = 𝑎)) → ((𝑎𝑉𝑏𝑉) → (𝑋𝑉𝑌𝑉)))
1716com12 32 . . . . . . . 8 ((𝑎𝑉𝑏𝑉) → (((𝑋 = 𝑎𝑌 = 𝑏) ∨ (𝑋 = 𝑏𝑌 = 𝑎)) → (𝑋𝑉𝑌𝑉)))
1817adantl 485 . . . . . . 7 (((𝑋𝑈𝑌𝑊) ∧ (𝑎𝑉𝑏𝑉)) → (((𝑋 = 𝑎𝑌 = 𝑏) ∨ (𝑋 = 𝑏𝑌 = 𝑎)) → (𝑋𝑉𝑌𝑉)))
192, 18sylbid 243 . . . . . 6 (((𝑋𝑈𝑌𝑊) ∧ (𝑎𝑉𝑏𝑉)) → ({𝑋, 𝑌} = {𝑎, 𝑏} → (𝑋𝑉𝑌𝑉)))
2019expcom 417 . . . . 5 ((𝑎𝑉𝑏𝑉) → ((𝑋𝑈𝑌𝑊) → ({𝑋, 𝑌} = {𝑎, 𝑏} → (𝑋𝑉𝑌𝑉))))
2120com23 86 . . . 4 ((𝑎𝑉𝑏𝑉) → ({𝑋, 𝑌} = {𝑎, 𝑏} → ((𝑋𝑈𝑌𝑊) → (𝑋𝑉𝑌𝑉))))
2221rexlimivv 3251 . . 3 (∃𝑎𝑉𝑏𝑉 {𝑋, 𝑌} = {𝑎, 𝑏} → ((𝑋𝑈𝑌𝑊) → (𝑋𝑉𝑌𝑉)))
231, 22syl 17 . 2 ({𝑋, 𝑌} ∈ (Pairs‘𝑉) → ((𝑋𝑈𝑌𝑊) → (𝑋𝑉𝑌𝑉)))
2423imp 410 1 (({𝑋, 𝑌} ∈ (Pairs‘𝑉) ∧ (𝑋𝑈𝑌𝑊)) → (𝑋𝑉𝑌𝑉))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111  ∃wrex 3107  {cpr 4527  ‘cfv 6325  Pairscspr 44037 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-spr 44038 This theorem is referenced by:  prsssprel  44048  sprsymrelfolem2  44053
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