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Theorem prsprel 45769
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 45766 . . 3 ({𝑋, 𝑌} ∈ (Pairs‘𝑉) → ∃𝑎𝑉𝑏𝑉 {𝑋, 𝑌} = {𝑎, 𝑏})
2 preq12bg 4815 . . . . . . 7 (((𝑋𝑈𝑌𝑊) ∧ (𝑎𝑉𝑏𝑉)) → ({𝑋, 𝑌} = {𝑎, 𝑏} ↔ ((𝑋 = 𝑎𝑌 = 𝑏) ∨ (𝑋 = 𝑏𝑌 = 𝑎))))
3 eleq1 2822 . . . . . . . . . . . . 13 (𝑎 = 𝑋 → (𝑎𝑉𝑋𝑉))
43eqcoms 2741 . . . . . . . . . . . 12 (𝑋 = 𝑎 → (𝑎𝑉𝑋𝑉))
5 eleq1 2822 . . . . . . . . . . . . 13 (𝑏 = 𝑌 → (𝑏𝑉𝑌𝑉))
65eqcoms 2741 . . . . . . . . . . . 12 (𝑌 = 𝑏 → (𝑏𝑉𝑌𝑉))
74, 6bi2anan9 638 . . . . . . . . . . 11 ((𝑋 = 𝑎𝑌 = 𝑏) → ((𝑎𝑉𝑏𝑉) ↔ (𝑋𝑉𝑌𝑉)))
87biimpd 228 . . . . . . . . . 10 ((𝑋 = 𝑎𝑌 = 𝑏) → ((𝑎𝑉𝑏𝑉) → (𝑋𝑉𝑌𝑉)))
9 eleq1 2822 . . . . . . . . . . . . . 14 (𝑏 = 𝑋 → (𝑏𝑉𝑋𝑉))
109eqcoms 2741 . . . . . . . . . . . . 13 (𝑋 = 𝑏 → (𝑏𝑉𝑋𝑉))
11 eleq1 2822 . . . . . . . . . . . . . 14 (𝑎 = 𝑌 → (𝑎𝑉𝑌𝑉))
1211eqcoms 2741 . . . . . . . . . . . . 13 (𝑌 = 𝑎 → (𝑎𝑉𝑌𝑉))
1310, 12bi2anan9 638 . . . . . . . . . . . 12 ((𝑋 = 𝑏𝑌 = 𝑎) → ((𝑏𝑉𝑎𝑉) ↔ (𝑋𝑉𝑌𝑉)))
1413biimpd 228 . . . . . . . . . . 11 ((𝑋 = 𝑏𝑌 = 𝑎) → ((𝑏𝑉𝑎𝑉) → (𝑋𝑉𝑌𝑉)))
1514ancomsd 467 . . . . . . . . . 10 ((𝑋 = 𝑏𝑌 = 𝑎) → ((𝑎𝑉𝑏𝑉) → (𝑋𝑉𝑌𝑉)))
168, 15jaoi 856 . . . . . . . . 9 (((𝑋 = 𝑎𝑌 = 𝑏) ∨ (𝑋 = 𝑏𝑌 = 𝑎)) → ((𝑎𝑉𝑏𝑉) → (𝑋𝑉𝑌𝑉)))
1716com12 32 . . . . . . . 8 ((𝑎𝑉𝑏𝑉) → (((𝑋 = 𝑎𝑌 = 𝑏) ∨ (𝑋 = 𝑏𝑌 = 𝑎)) → (𝑋𝑉𝑌𝑉)))
1817adantl 483 . . . . . . 7 (((𝑋𝑈𝑌𝑊) ∧ (𝑎𝑉𝑏𝑉)) → (((𝑋 = 𝑎𝑌 = 𝑏) ∨ (𝑋 = 𝑏𝑌 = 𝑎)) → (𝑋𝑉𝑌𝑉)))
192, 18sylbid 239 . . . . . 6 (((𝑋𝑈𝑌𝑊) ∧ (𝑎𝑉𝑏𝑉)) → ({𝑋, 𝑌} = {𝑎, 𝑏} → (𝑋𝑉𝑌𝑉)))
2019expcom 415 . . . . 5 ((𝑎𝑉𝑏𝑉) → ((𝑋𝑈𝑌𝑊) → ({𝑋, 𝑌} = {𝑎, 𝑏} → (𝑋𝑉𝑌𝑉))))
2120com23 86 . . . 4 ((𝑎𝑉𝑏𝑉) → ({𝑋, 𝑌} = {𝑎, 𝑏} → ((𝑋𝑈𝑌𝑊) → (𝑋𝑉𝑌𝑉))))
2221rexlimivv 3193 . . 3 (∃𝑎𝑉𝑏𝑉 {𝑋, 𝑌} = {𝑎, 𝑏} → ((𝑋𝑈𝑌𝑊) → (𝑋𝑉𝑌𝑉)))
231, 22syl 17 . 2 ({𝑋, 𝑌} ∈ (Pairs‘𝑉) → ((𝑋𝑈𝑌𝑊) → (𝑋𝑉𝑌𝑉)))
2423imp 408 1 (({𝑋, 𝑌} ∈ (Pairs‘𝑉) ∧ (𝑋𝑈𝑌𝑊)) → (𝑋𝑉𝑌𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wo 846   = wceq 1542  wcel 2107  wrex 3070  {cpr 4592  cfv 6500  Pairscspr 45759
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-spr 45760
This theorem is referenced by:  prsssprel  45770  sprsymrelfolem2  45775
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