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Theorem prprelprb 46185
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 46183 . . . . 5 (𝑋 ∈ V → (Pairsproper𝑋) = {𝑝 ∈ 𝒫 𝑋 ∣ ∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑝 = {𝑎, 𝑏})})
21eleq2d 2820 . . . 4 (𝑋 ∈ V → (𝑃 ∈ (Pairsproper𝑋) ↔ 𝑃 ∈ {𝑝 ∈ 𝒫 𝑋 ∣ ∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑝 = {𝑎, 𝑏})}))
3 eqeq1 2737 . . . . . . 7 (𝑝 = 𝑃 → (𝑝 = {𝑎, 𝑏} ↔ 𝑃 = {𝑎, 𝑏}))
43anbi2d 630 . . . . . 6 (𝑝 = 𝑃 → ((𝑎𝑏𝑝 = {𝑎, 𝑏}) ↔ (𝑎𝑏𝑃 = {𝑎, 𝑏})))
542rexbidv 3220 . . . . 5 (𝑝 = 𝑃 → (∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑝 = {𝑎, 𝑏}) ↔ ∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑃 = {𝑎, 𝑏})))
65elrab 3684 . . . 4 (𝑃 ∈ {𝑝 ∈ 𝒫 𝑋 ∣ ∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑝 = {𝑎, 𝑏})} ↔ (𝑃 ∈ 𝒫 𝑋 ∧ ∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑃 = {𝑎, 𝑏})))
72, 6bitrdi 287 . . 3 (𝑋 ∈ V → (𝑃 ∈ (Pairsproper𝑋) ↔ (𝑃 ∈ 𝒫 𝑋 ∧ ∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑃 = {𝑎, 𝑏}))))
8 pm3.22 461 . . . . . . . . 9 ((𝑎𝑏𝑃 = {𝑎, 𝑏}) → (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏))
98a1i 11 . . . . . . . 8 ((𝑃 ∈ 𝒫 𝑋 ∧ (𝑎𝑋𝑏𝑋)) → ((𝑎𝑏𝑃 = {𝑎, 𝑏}) → (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏)))
109reximdvva 3206 . . . . . . 7 (𝑃 ∈ 𝒫 𝑋 → (∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑃 = {𝑎, 𝑏}) → ∃𝑎𝑋𝑏𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏)))
1110imp 408 . . . . . 6 ((𝑃 ∈ 𝒫 𝑋 ∧ ∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑃 = {𝑎, 𝑏})) → ∃𝑎𝑋𝑏𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏))
1211anim2i 618 . . . . 5 ((𝑋 ∈ V ∧ (𝑃 ∈ 𝒫 𝑋 ∧ ∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑃 = {𝑎, 𝑏}))) → (𝑋 ∈ V ∧ ∃𝑎𝑋𝑏𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏)))
1312ex 414 . . . 4 (𝑋 ∈ V → ((𝑃 ∈ 𝒫 𝑋 ∧ ∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑃 = {𝑎, 𝑏})) → (𝑋 ∈ V ∧ ∃𝑎𝑋𝑏𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏))))
14 simpr 486 . . . . . . . . . 10 (((𝑋 ∈ V ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏)) → (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏))
1514ancomd 463 . . . . . . . . 9 (((𝑋 ∈ V ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏)) → (𝑎𝑏𝑃 = {𝑎, 𝑏}))
16 prelpwi 5448 . . . . . . . . . . . 12 ((𝑎𝑋𝑏𝑋) → {𝑎, 𝑏} ∈ 𝒫 𝑋)
1716adantl 483 . . . . . . . . . . 11 ((𝑋 ∈ V ∧ (𝑎𝑋𝑏𝑋)) → {𝑎, 𝑏} ∈ 𝒫 𝑋)
1817adantr 482 . . . . . . . . . 10 (((𝑋 ∈ V ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏)) → {𝑎, 𝑏} ∈ 𝒫 𝑋)
19 eleq1 2822 . . . . . . . . . . . 12 (𝑃 = {𝑎, 𝑏} → (𝑃 ∈ 𝒫 𝑋 ↔ {𝑎, 𝑏} ∈ 𝒫 𝑋))
2019adantr 482 . . . . . . . . . . 11 ((𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏) → (𝑃 ∈ 𝒫 𝑋 ↔ {𝑎, 𝑏} ∈ 𝒫 𝑋))
2120adantl 483 . . . . . . . . . 10 (((𝑋 ∈ V ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏)) → (𝑃 ∈ 𝒫 𝑋 ↔ {𝑎, 𝑏} ∈ 𝒫 𝑋))
2218, 21mpbird 257 . . . . . . . . 9 (((𝑋 ∈ V ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏)) → 𝑃 ∈ 𝒫 𝑋)
2315, 22jca 513 . . . . . . . 8 (((𝑋 ∈ V ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏)) → ((𝑎𝑏𝑃 = {𝑎, 𝑏}) ∧ 𝑃 ∈ 𝒫 𝑋))
2423ex 414 . . . . . . 7 ((𝑋 ∈ V ∧ (𝑎𝑋𝑏𝑋)) → ((𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏) → ((𝑎𝑏𝑃 = {𝑎, 𝑏}) ∧ 𝑃 ∈ 𝒫 𝑋)))
2524reximdvva 3206 . . . . . 6 (𝑋 ∈ V → (∃𝑎𝑋𝑏𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏) → ∃𝑎𝑋𝑏𝑋 ((𝑎𝑏𝑃 = {𝑎, 𝑏}) ∧ 𝑃 ∈ 𝒫 𝑋)))
2625imp 408 . . . . 5 ((𝑋 ∈ V ∧ ∃𝑎𝑋𝑏𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏)) → ∃𝑎𝑋𝑏𝑋 ((𝑎𝑏𝑃 = {𝑎, 𝑏}) ∧ 𝑃 ∈ 𝒫 𝑋))
27 r19.41vv 3225 . . . . . 6 (∃𝑎𝑋𝑏𝑋 ((𝑎𝑏𝑃 = {𝑎, 𝑏}) ∧ 𝑃 ∈ 𝒫 𝑋) ↔ (∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑃 = {𝑎, 𝑏}) ∧ 𝑃 ∈ 𝒫 𝑋))
2827biancomi 464 . . . . 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 279 . 2 (𝑋 ∈ V → (𝑃 ∈ (Pairsproper𝑋) ↔ (𝑋 ∈ V ∧ ∃𝑎𝑋𝑏𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏))))
32 fvprc 6884 . . . 4 𝑋 ∈ V → (Pairsproper𝑋) = ∅)
3332eleq2d 2820 . . 3 𝑋 ∈ V → (𝑃 ∈ (Pairsproper𝑋) ↔ 𝑃 ∈ ∅))
34 noel 4331 . . . . 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 412 . . . 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 279 . 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 397   = wceq 1542  wcel 2107  wne 2941  wrex 3071  {crab 3433  Vcvv 3475  c0 4323  𝒫 cpw 4603  {cpr 4631  cfv 6544  Pairspropercprpr 46180
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-prpr 46181
This theorem is referenced by:  inlinecirc02p  47473
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