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Theorem prprvalpw 47440
Description: The set of all proper unordered pairs over a given set 𝑉, expressed by a restricted class abstraction. (Contributed by AV, 29-Apr-2023.)
Assertion
Ref Expression
prprvalpw (𝑉𝑊 → (Pairsproper𝑉) = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})})
Distinct variable groups:   𝑉,𝑎,𝑏,𝑝   𝑊,𝑎,𝑏,𝑝

Proof of Theorem prprvalpw
StepHypRef Expression
1 prprval 47439 . 2 (𝑉𝑊 → (Pairsproper𝑉) = {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})})
2 prssi 4826 . . . . . . . 8 ((𝑎𝑉𝑏𝑉) → {𝑎, 𝑏} ⊆ 𝑉)
3 eleq1 2827 . . . . . . . . . 10 (𝑝 = {𝑎, 𝑏} → (𝑝 ∈ 𝒫 𝑉 ↔ {𝑎, 𝑏} ∈ 𝒫 𝑉))
43adantl 481 . . . . . . . . 9 ((𝑎𝑏𝑝 = {𝑎, 𝑏}) → (𝑝 ∈ 𝒫 𝑉 ↔ {𝑎, 𝑏} ∈ 𝒫 𝑉))
5 prex 5443 . . . . . . . . . 10 {𝑎, 𝑏} ∈ V
65elpw 4609 . . . . . . . . 9 ({𝑎, 𝑏} ∈ 𝒫 𝑉 ↔ {𝑎, 𝑏} ⊆ 𝑉)
74, 6bitrdi 287 . . . . . . . 8 ((𝑎𝑏𝑝 = {𝑎, 𝑏}) → (𝑝 ∈ 𝒫 𝑉 ↔ {𝑎, 𝑏} ⊆ 𝑉))
82, 7syl5ibrcom 247 . . . . . . 7 ((𝑎𝑉𝑏𝑉) → ((𝑎𝑏𝑝 = {𝑎, 𝑏}) → 𝑝 ∈ 𝒫 𝑉))
98rexlimivv 3199 . . . . . 6 (∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏}) → 𝑝 ∈ 𝒫 𝑉)
109pm4.71ri 560 . . . . 5 (∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏}) ↔ (𝑝 ∈ 𝒫 𝑉 ∧ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})))
1110a1i 11 . . . 4 (𝑉𝑊 → (∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏}) ↔ (𝑝 ∈ 𝒫 𝑉 ∧ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏}))))
1211abbidv 2806 . . 3 (𝑉𝑊 → {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})} = {𝑝 ∣ (𝑝 ∈ 𝒫 𝑉 ∧ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏}))})
13 df-rab 3434 . . 3 {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})} = {𝑝 ∣ (𝑝 ∈ 𝒫 𝑉 ∧ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏}))}
1412, 13eqtr4di 2793 . 2 (𝑉𝑊 → {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})} = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})})
151, 14eqtrd 2775 1 (𝑉𝑊 → (Pairsproper𝑉) = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  {cab 2712  wne 2938  wrex 3068  {crab 3433  wss 3963  𝒫 cpw 4605  {cpr 4633  cfv 6563  Pairspropercprpr 47437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-prpr 47438
This theorem is referenced by:  prprelb  47441  prprelprb  47442
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