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Theorem prprelprb 46264
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 46262 . . . . 5 (𝑋 ∈ V → (Pairsproper𝑋) = {𝑝 ∈ 𝒫 𝑋 ∣ ∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑝 = {𝑎, 𝑏})})
21eleq2d 2819 . . . 4 (𝑋 ∈ V → (𝑃 ∈ (Pairsproper𝑋) ↔ 𝑃 ∈ {𝑝 ∈ 𝒫 𝑋 ∣ ∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑝 = {𝑎, 𝑏})}))
3 eqeq1 2736 . . . . . . 7 (𝑝 = 𝑃 → (𝑝 = {𝑎, 𝑏} ↔ 𝑃 = {𝑎, 𝑏}))
43anbi2d 629 . . . . . 6 (𝑝 = 𝑃 → ((𝑎𝑏𝑝 = {𝑎, 𝑏}) ↔ (𝑎𝑏𝑃 = {𝑎, 𝑏})))
542rexbidv 3219 . . . . 5 (𝑝 = 𝑃 → (∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑝 = {𝑎, 𝑏}) ↔ ∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑃 = {𝑎, 𝑏})))
65elrab 3683 . . . 4 (𝑃 ∈ {𝑝 ∈ 𝒫 𝑋 ∣ ∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑝 = {𝑎, 𝑏})} ↔ (𝑃 ∈ 𝒫 𝑋 ∧ ∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑃 = {𝑎, 𝑏})))
72, 6bitrdi 286 . . 3 (𝑋 ∈ V → (𝑃 ∈ (Pairsproper𝑋) ↔ (𝑃 ∈ 𝒫 𝑋 ∧ ∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑃 = {𝑎, 𝑏}))))
8 pm3.22 460 . . . . . . . . 9 ((𝑎𝑏𝑃 = {𝑎, 𝑏}) → (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏))
98a1i 11 . . . . . . . 8 ((𝑃 ∈ 𝒫 𝑋 ∧ (𝑎𝑋𝑏𝑋)) → ((𝑎𝑏𝑃 = {𝑎, 𝑏}) → (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏)))
109reximdvva 3205 . . . . . . 7 (𝑃 ∈ 𝒫 𝑋 → (∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑃 = {𝑎, 𝑏}) → ∃𝑎𝑋𝑏𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏)))
1110imp 407 . . . . . 6 ((𝑃 ∈ 𝒫 𝑋 ∧ ∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑃 = {𝑎, 𝑏})) → ∃𝑎𝑋𝑏𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏))
1211anim2i 617 . . . . 5 ((𝑋 ∈ V ∧ (𝑃 ∈ 𝒫 𝑋 ∧ ∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑃 = {𝑎, 𝑏}))) → (𝑋 ∈ V ∧ ∃𝑎𝑋𝑏𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏)))
1312ex 413 . . . 4 (𝑋 ∈ V → ((𝑃 ∈ 𝒫 𝑋 ∧ ∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑃 = {𝑎, 𝑏})) → (𝑋 ∈ V ∧ ∃𝑎𝑋𝑏𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏))))
14 simpr 485 . . . . . . . . . 10 (((𝑋 ∈ V ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏)) → (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏))
1514ancomd 462 . . . . . . . . 9 (((𝑋 ∈ V ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏)) → (𝑎𝑏𝑃 = {𝑎, 𝑏}))
16 prelpwi 5447 . . . . . . . . . . . 12 ((𝑎𝑋𝑏𝑋) → {𝑎, 𝑏} ∈ 𝒫 𝑋)
1716adantl 482 . . . . . . . . . . 11 ((𝑋 ∈ V ∧ (𝑎𝑋𝑏𝑋)) → {𝑎, 𝑏} ∈ 𝒫 𝑋)
1817adantr 481 . . . . . . . . . 10 (((𝑋 ∈ V ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏)) → {𝑎, 𝑏} ∈ 𝒫 𝑋)
19 eleq1 2821 . . . . . . . . . . . 12 (𝑃 = {𝑎, 𝑏} → (𝑃 ∈ 𝒫 𝑋 ↔ {𝑎, 𝑏} ∈ 𝒫 𝑋))
2019adantr 481 . . . . . . . . . . 11 ((𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏) → (𝑃 ∈ 𝒫 𝑋 ↔ {𝑎, 𝑏} ∈ 𝒫 𝑋))
2120adantl 482 . . . . . . . . . 10 (((𝑋 ∈ V ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏)) → (𝑃 ∈ 𝒫 𝑋 ↔ {𝑎, 𝑏} ∈ 𝒫 𝑋))
2218, 21mpbird 256 . . . . . . . . 9 (((𝑋 ∈ V ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏)) → 𝑃 ∈ 𝒫 𝑋)
2315, 22jca 512 . . . . . . . 8 (((𝑋 ∈ V ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏)) → ((𝑎𝑏𝑃 = {𝑎, 𝑏}) ∧ 𝑃 ∈ 𝒫 𝑋))
2423ex 413 . . . . . . 7 ((𝑋 ∈ V ∧ (𝑎𝑋𝑏𝑋)) → ((𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏) → ((𝑎𝑏𝑃 = {𝑎, 𝑏}) ∧ 𝑃 ∈ 𝒫 𝑋)))
2524reximdvva 3205 . . . . . 6 (𝑋 ∈ V → (∃𝑎𝑋𝑏𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏) → ∃𝑎𝑋𝑏𝑋 ((𝑎𝑏𝑃 = {𝑎, 𝑏}) ∧ 𝑃 ∈ 𝒫 𝑋)))
2625imp 407 . . . . 5 ((𝑋 ∈ V ∧ ∃𝑎𝑋𝑏𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏)) → ∃𝑎𝑋𝑏𝑋 ((𝑎𝑏𝑃 = {𝑎, 𝑏}) ∧ 𝑃 ∈ 𝒫 𝑋))
27 r19.41vv 3224 . . . . . 6 (∃𝑎𝑋𝑏𝑋 ((𝑎𝑏𝑃 = {𝑎, 𝑏}) ∧ 𝑃 ∈ 𝒫 𝑋) ↔ (∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑃 = {𝑎, 𝑏}) ∧ 𝑃 ∈ 𝒫 𝑋))
2827biancomi 463 . . . . 5 (∃𝑎𝑋𝑏𝑋 ((𝑎𝑏𝑃 = {𝑎, 𝑏}) ∧ 𝑃 ∈ 𝒫 𝑋) ↔ (𝑃 ∈ 𝒫 𝑋 ∧ ∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑃 = {𝑎, 𝑏})))
2926, 28sylib 217 . . . 4 ((𝑋 ∈ V ∧ ∃𝑎𝑋𝑏𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏)) → (𝑃 ∈ 𝒫 𝑋 ∧ ∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑃 = {𝑎, 𝑏})))
3013, 29impbid1 224 . . 3 (𝑋 ∈ V → ((𝑃 ∈ 𝒫 𝑋 ∧ ∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑃 = {𝑎, 𝑏})) ↔ (𝑋 ∈ V ∧ ∃𝑎𝑋𝑏𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏))))
317, 30bitrd 278 . 2 (𝑋 ∈ V → (𝑃 ∈ (Pairsproper𝑋) ↔ (𝑋 ∈ V ∧ ∃𝑎𝑋𝑏𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏))))
32 fvprc 6883 . . . 4 𝑋 ∈ V → (Pairsproper𝑋) = ∅)
3332eleq2d 2819 . . 3 𝑋 ∈ V → (𝑃 ∈ (Pairsproper𝑋) ↔ 𝑃 ∈ ∅))
34 noel 4330 . . . . 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 411 . . . 4 𝑋 ∈ V → ((𝑋 ∈ V ∧ ∃𝑎𝑋𝑏𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏)) → 𝑃 ∈ ∅))
3936, 38impbid 211 . . 3 𝑋 ∈ V → (𝑃 ∈ ∅ ↔ (𝑋 ∈ V ∧ ∃𝑎𝑋𝑏𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏))))
4033, 39bitrd 278 . 2 𝑋 ∈ V → (𝑃 ∈ (Pairsproper𝑋) ↔ (𝑋 ∈ V ∧ ∃𝑎𝑋𝑏𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏))))
4131, 40pm2.61i 182 1 (𝑃 ∈ (Pairsproper𝑋) ↔ (𝑋 ∈ V ∧ ∃𝑎𝑋𝑏𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wne 2940  wrex 3070  {crab 3432  Vcvv 3474  c0 4322  𝒫 cpw 4602  {cpr 4630  cfv 6543  Pairspropercprpr 46259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-prpr 46260
This theorem is referenced by:  inlinecirc02p  47551
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