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Theorem prssspr 47472
Description: An element of a subset of the set of all unordered pairs over a given set 𝑉, is a pair of elements of the set 𝑉. (Contributed by AV, 22-Nov-2021.)
Assertion
Ref Expression
prssspr ((𝑃 ⊆ (Pairs‘𝑉) ∧ 𝑋𝑃) → ∃𝑎𝑉𝑏𝑉 𝑋 = {𝑎, 𝑏})
Distinct variable groups:   𝑉,𝑎,𝑏   𝑋,𝑎,𝑏
Allowed substitution hints:   𝑃(𝑎,𝑏)

Proof of Theorem prssspr
StepHypRef Expression
1 ssel2 3978 . 2 ((𝑃 ⊆ (Pairs‘𝑉) ∧ 𝑋𝑃) → 𝑋 ∈ (Pairs‘𝑉))
2 sprel 47471 . 2 (𝑋 ∈ (Pairs‘𝑉) → ∃𝑎𝑉𝑏𝑉 𝑋 = {𝑎, 𝑏})
31, 2syl 17 1 ((𝑃 ⊆ (Pairs‘𝑉) ∧ 𝑋𝑃) → ∃𝑎𝑉𝑏𝑉 𝑋 = {𝑎, 𝑏})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wrex 3070  wss 3951  {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:  sprsymrelf1lem  47478
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