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Theorem prelspr 47473
Description: An unordered pair of elements of a fixed set 𝑉 belongs to the set of all unordered pairs over the set 𝑉. (Contributed by AV, 21-Nov-2021.)
Assertion
Ref Expression
prelspr ((𝑉𝑊 ∧ (𝑋𝑉𝑌𝑉)) → {𝑋, 𝑌} ∈ (Pairs‘𝑉))

Proof of Theorem prelspr
Dummy variables 𝑎 𝑏 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prelpwi 5452 . . . . 5 ((𝑋𝑉𝑌𝑉) → {𝑋, 𝑌} ∈ 𝒫 𝑉)
2 eqidd 2738 . . . . . 6 ((𝑋𝑉𝑌𝑉) → {𝑋, 𝑌} = {𝑋, 𝑌})
3 preq1 4733 . . . . . . . 8 (𝑎 = 𝑋 → {𝑎, 𝑏} = {𝑋, 𝑏})
43eqeq2d 2748 . . . . . . 7 (𝑎 = 𝑋 → ({𝑋, 𝑌} = {𝑎, 𝑏} ↔ {𝑋, 𝑌} = {𝑋, 𝑏}))
5 preq2 4734 . . . . . . . 8 (𝑏 = 𝑌 → {𝑋, 𝑏} = {𝑋, 𝑌})
65eqeq2d 2748 . . . . . . 7 (𝑏 = 𝑌 → ({𝑋, 𝑌} = {𝑋, 𝑏} ↔ {𝑋, 𝑌} = {𝑋, 𝑌}))
74, 6rspc2ev 3635 . . . . . 6 ((𝑋𝑉𝑌𝑉 ∧ {𝑋, 𝑌} = {𝑋, 𝑌}) → ∃𝑎𝑉𝑏𝑉 {𝑋, 𝑌} = {𝑎, 𝑏})
82, 7mpd3an3 1464 . . . . 5 ((𝑋𝑉𝑌𝑉) → ∃𝑎𝑉𝑏𝑉 {𝑋, 𝑌} = {𝑎, 𝑏})
91, 8jca 511 . . . 4 ((𝑋𝑉𝑌𝑉) → ({𝑋, 𝑌} ∈ 𝒫 𝑉 ∧ ∃𝑎𝑉𝑏𝑉 {𝑋, 𝑌} = {𝑎, 𝑏}))
109adantl 481 . . 3 ((𝑉𝑊 ∧ (𝑋𝑉𝑌𝑉)) → ({𝑋, 𝑌} ∈ 𝒫 𝑉 ∧ ∃𝑎𝑉𝑏𝑉 {𝑋, 𝑌} = {𝑎, 𝑏}))
11 eqeq1 2741 . . . . 5 (𝑝 = {𝑋, 𝑌} → (𝑝 = {𝑎, 𝑏} ↔ {𝑋, 𝑌} = {𝑎, 𝑏}))
12112rexbidv 3222 . . . 4 (𝑝 = {𝑋, 𝑌} → (∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎𝑉𝑏𝑉 {𝑋, 𝑌} = {𝑎, 𝑏}))
1312elrab 3692 . . 3 ({𝑋, 𝑌} ∈ {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}} ↔ ({𝑋, 𝑌} ∈ 𝒫 𝑉 ∧ ∃𝑎𝑉𝑏𝑉 {𝑋, 𝑌} = {𝑎, 𝑏}))
1410, 13sylibr 234 . 2 ((𝑉𝑊 ∧ (𝑋𝑉𝑌𝑉)) → {𝑋, 𝑌} ∈ {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})
15 sprvalpw 47467 . . 3 (𝑉𝑊 → (Pairs‘𝑉) = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})
1615adantr 480 . 2 ((𝑉𝑊 ∧ (𝑋𝑉𝑌𝑉)) → (Pairs‘𝑉) = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})
1714, 16eleqtrrd 2844 1 ((𝑉𝑊 ∧ (𝑋𝑉𝑌𝑉)) → {𝑋, 𝑌} ∈ (Pairs‘𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wrex 3070  {crab 3436  𝒫 cpw 4600  {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:  sprsymrelfolem2  47480  reupr  47509
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