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Theorem prprelprb 44202
 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 44200 . . . . 5 (𝑋 ∈ V → (Pairsproper𝑋) = {𝑝 ∈ 𝒫 𝑋 ∣ ∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑝 = {𝑎, 𝑏})})
21eleq2d 2875 . . . 4 (𝑋 ∈ V → (𝑃 ∈ (Pairsproper𝑋) ↔ 𝑃 ∈ {𝑝 ∈ 𝒫 𝑋 ∣ ∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑝 = {𝑎, 𝑏})}))
3 eqeq1 2802 . . . . . . 7 (𝑝 = 𝑃 → (𝑝 = {𝑎, 𝑏} ↔ 𝑃 = {𝑎, 𝑏}))
43anbi2d 631 . . . . . 6 (𝑝 = 𝑃 → ((𝑎𝑏𝑝 = {𝑎, 𝑏}) ↔ (𝑎𝑏𝑃 = {𝑎, 𝑏})))
542rexbidv 3260 . . . . 5 (𝑝 = 𝑃 → (∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑝 = {𝑎, 𝑏}) ↔ ∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑃 = {𝑎, 𝑏})))
65elrab 3630 . . . 4 (𝑃 ∈ {𝑝 ∈ 𝒫 𝑋 ∣ ∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑝 = {𝑎, 𝑏})} ↔ (𝑃 ∈ 𝒫 𝑋 ∧ ∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝑃 = {𝑎, 𝑏})))
72, 6syl6bb 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 3237 . . . . . . 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 619 . . . . 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 5309 . . . . . . . . . . . 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 2877 . . . . . . . . . . . 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 3237 . . . . . 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 3303 . . . . . 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 6647 . . . 4 𝑋 ∈ V → (Pairsproper𝑋) = ∅)
3332eleq2d 2875 . . 3 𝑋 ∈ V → (𝑃 ∈ (Pairsproper𝑋) ↔ 𝑃 ∈ ∅))
34 noel 4250 . . . . 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 1538   ∈ wcel 2111   ≠ wne 2987  ∃wrex 3107  {crab 3110  Vcvv 3442  ∅c0 4246  𝒫 cpw 4500  {cpr 4530  ‘cfv 6332  Pairspropercprpr 44197 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pr 5299  ax-un 7454 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-prpr 44198 This theorem is referenced by:  inlinecirc02p  45367
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