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Theorem prprelprb 44642
Description: A set is an element of the set of all proper unordered pairs over a given set 𝑋 iff it is a pair of different elements of the set 𝑋. (Contributed by AV, 7-May-2023.)
Assertion
Ref Expression
prprelprb (𝑃 ∈ (Pairsproper𝑋) ↔ (𝑋 ∈ V ∧ ∃𝑎𝑋𝑏𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏)))
Distinct variable groups:   𝑃,𝑎,𝑏   𝑋,𝑎,𝑏

Proof of Theorem prprelprb
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 prprvalpw 44640 . . . . 5 (𝑋 ∈ V → (Pairsproper𝑋) = {𝑝 ∈ 𝒫 𝑋 ∣ ∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑝 = {𝑎, 𝑏})})
21eleq2d 2823 . . . 4 (𝑋 ∈ V → (𝑃 ∈ (Pairsproper𝑋) ↔ 𝑃 ∈ {𝑝 ∈ 𝒫 𝑋 ∣ ∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑝 = {𝑎, 𝑏})}))
3 eqeq1 2741 . . . . . . 7 (𝑝 = 𝑃 → (𝑝 = {𝑎, 𝑏} ↔ 𝑃 = {𝑎, 𝑏}))
43anbi2d 632 . . . . . 6 (𝑝 = 𝑃 → ((𝑎𝑏𝑝 = {𝑎, 𝑏}) ↔ (𝑎𝑏𝑃 = {𝑎, 𝑏})))
542rexbidv 3219 . . . . 5 (𝑝 = 𝑃 → (∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑝 = {𝑎, 𝑏}) ↔ ∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑃 = {𝑎, 𝑏})))
65elrab 3602 . . . 4 (𝑃 ∈ {𝑝 ∈ 𝒫 𝑋 ∣ ∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑝 = {𝑎, 𝑏})} ↔ (𝑃 ∈ 𝒫 𝑋 ∧ ∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑃 = {𝑎, 𝑏})))
72, 6bitrdi 290 . . 3 (𝑋 ∈ V → (𝑃 ∈ (Pairsproper𝑋) ↔ (𝑃 ∈ 𝒫 𝑋 ∧ ∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑃 = {𝑎, 𝑏}))))
8 pm3.22 463 . . . . . . . . 9 ((𝑎𝑏𝑃 = {𝑎, 𝑏}) → (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏))
98a1i 11 . . . . . . . 8 ((𝑃 ∈ 𝒫 𝑋 ∧ (𝑎𝑋𝑏𝑋)) → ((𝑎𝑏𝑃 = {𝑎, 𝑏}) → (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏)))
109reximdvva 3196 . . . . . . 7 (𝑃 ∈ 𝒫 𝑋 → (∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑃 = {𝑎, 𝑏}) → ∃𝑎𝑋𝑏𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏)))
1110imp 410 . . . . . 6 ((𝑃 ∈ 𝒫 𝑋 ∧ ∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑃 = {𝑎, 𝑏})) → ∃𝑎𝑋𝑏𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏))
1211anim2i 620 . . . . 5 ((𝑋 ∈ V ∧ (𝑃 ∈ 𝒫 𝑋 ∧ ∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑃 = {𝑎, 𝑏}))) → (𝑋 ∈ V ∧ ∃𝑎𝑋𝑏𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏)))
1312ex 416 . . . 4 (𝑋 ∈ V → ((𝑃 ∈ 𝒫 𝑋 ∧ ∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑃 = {𝑎, 𝑏})) → (𝑋 ∈ V ∧ ∃𝑎𝑋𝑏𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏))))
14 simpr 488 . . . . . . . . . 10 (((𝑋 ∈ V ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏)) → (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏))
1514ancomd 465 . . . . . . . . 9 (((𝑋 ∈ V ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏)) → (𝑎𝑏𝑃 = {𝑎, 𝑏}))
16 prelpwi 5332 . . . . . . . . . . . 12 ((𝑎𝑋𝑏𝑋) → {𝑎, 𝑏} ∈ 𝒫 𝑋)
1716adantl 485 . . . . . . . . . . 11 ((𝑋 ∈ V ∧ (𝑎𝑋𝑏𝑋)) → {𝑎, 𝑏} ∈ 𝒫 𝑋)
1817adantr 484 . . . . . . . . . 10 (((𝑋 ∈ V ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏)) → {𝑎, 𝑏} ∈ 𝒫 𝑋)
19 eleq1 2825 . . . . . . . . . . . 12 (𝑃 = {𝑎, 𝑏} → (𝑃 ∈ 𝒫 𝑋 ↔ {𝑎, 𝑏} ∈ 𝒫 𝑋))
2019adantr 484 . . . . . . . . . . 11 ((𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏) → (𝑃 ∈ 𝒫 𝑋 ↔ {𝑎, 𝑏} ∈ 𝒫 𝑋))
2120adantl 485 . . . . . . . . . 10 (((𝑋 ∈ V ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏)) → (𝑃 ∈ 𝒫 𝑋 ↔ {𝑎, 𝑏} ∈ 𝒫 𝑋))
2218, 21mpbird 260 . . . . . . . . 9 (((𝑋 ∈ V ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏)) → 𝑃 ∈ 𝒫 𝑋)
2315, 22jca 515 . . . . . . . 8 (((𝑋 ∈ V ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏)) → ((𝑎𝑏𝑃 = {𝑎, 𝑏}) ∧ 𝑃 ∈ 𝒫 𝑋))
2423ex 416 . . . . . . 7 ((𝑋 ∈ V ∧ (𝑎𝑋𝑏𝑋)) → ((𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏) → ((𝑎𝑏𝑃 = {𝑎, 𝑏}) ∧ 𝑃 ∈ 𝒫 𝑋)))
2524reximdvva 3196 . . . . . 6 (𝑋 ∈ V → (∃𝑎𝑋𝑏𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏) → ∃𝑎𝑋𝑏𝑋 ((𝑎𝑏𝑃 = {𝑎, 𝑏}) ∧ 𝑃 ∈ 𝒫 𝑋)))
2625imp 410 . . . . 5 ((𝑋 ∈ V ∧ ∃𝑎𝑋𝑏𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏)) → ∃𝑎𝑋𝑏𝑋 ((𝑎𝑏𝑃 = {𝑎, 𝑏}) ∧ 𝑃 ∈ 𝒫 𝑋))
27 r19.41vv 3262 . . . . . 6 (∃𝑎𝑋𝑏𝑋 ((𝑎𝑏𝑃 = {𝑎, 𝑏}) ∧ 𝑃 ∈ 𝒫 𝑋) ↔ (∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑃 = {𝑎, 𝑏}) ∧ 𝑃 ∈ 𝒫 𝑋))
2827biancomi 466 . . . . 5 (∃𝑎𝑋𝑏𝑋 ((𝑎𝑏𝑃 = {𝑎, 𝑏}) ∧ 𝑃 ∈ 𝒫 𝑋) ↔ (𝑃 ∈ 𝒫 𝑋 ∧ ∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑃 = {𝑎, 𝑏})))
2926, 28sylib 221 . . . 4 ((𝑋 ∈ V ∧ ∃𝑎𝑋𝑏𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏)) → (𝑃 ∈ 𝒫 𝑋 ∧ ∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑃 = {𝑎, 𝑏})))
3013, 29impbid1 228 . . 3 (𝑋 ∈ V → ((𝑃 ∈ 𝒫 𝑋 ∧ ∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑃 = {𝑎, 𝑏})) ↔ (𝑋 ∈ V ∧ ∃𝑎𝑋𝑏𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏))))
317, 30bitrd 282 . 2 (𝑋 ∈ V → (𝑃 ∈ (Pairsproper𝑋) ↔ (𝑋 ∈ V ∧ ∃𝑎𝑋𝑏𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏))))
32 fvprc 6709 . . . 4 𝑋 ∈ V → (Pairsproper𝑋) = ∅)
3332eleq2d 2823 . . 3 𝑋 ∈ V → (𝑃 ∈ (Pairsproper𝑋) ↔ 𝑃 ∈ ∅))
34 noel 4245 . . . . 5 ¬ 𝑃 ∈ ∅
35 pm2.21 123 . . . . 5 𝑃 ∈ ∅ → (𝑃 ∈ ∅ → (𝑋 ∈ V ∧ ∃𝑎𝑋𝑏𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏))))
3634, 35mp1i 13 . . . 4 𝑋 ∈ V → (𝑃 ∈ ∅ → (𝑋 ∈ V ∧ ∃𝑎𝑋𝑏𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏))))
37 pm2.21 123 . . . . 5 𝑋 ∈ V → (𝑋 ∈ V → (∃𝑎𝑋𝑏𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏) → 𝑃 ∈ ∅)))
3837impd 414 . . . 4 𝑋 ∈ V → ((𝑋 ∈ V ∧ ∃𝑎𝑋𝑏𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏)) → 𝑃 ∈ ∅))
3936, 38impbid 215 . . 3 𝑋 ∈ V → (𝑃 ∈ ∅ ↔ (𝑋 ∈ V ∧ ∃𝑎𝑋𝑏𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏))))
4033, 39bitrd 282 . 2 𝑋 ∈ V → (𝑃 ∈ (Pairsproper𝑋) ↔ (𝑋 ∈ V ∧ ∃𝑎𝑋𝑏𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏))))
4131, 40pm2.61i 185 1 (𝑃 ∈ (Pairsproper𝑋) ↔ (𝑋 ∈ V ∧ ∃𝑎𝑋𝑏𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1543  wcel 2110  wne 2940  wrex 3062  {crab 3065  Vcvv 3408  c0 4237  𝒫 cpw 4513  {cpr 4543  cfv 6380  Pairspropercprpr 44637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-prpr 44638
This theorem is referenced by:  inlinecirc02p  45806
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