Theorem prprelprb 44969

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 ∈ (Pairsproper‘𝑋) ↔ (𝑋 ∈ V ∧ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)))

Distinct variable groups:   𝑃,𝑎,𝑏   𝑋,𝑎,𝑏

Proof of Theorem prprelprb

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		prprvalpw 44967	. . . . 5 ⊢ (𝑋 ∈ V → (Pairsproper‘𝑋) = {𝑝 ∈ 𝒫 𝑋 ∣ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})})
2	1	eleq2d 2824	. . . 4 ⊢ (𝑋 ∈ V → (𝑃 ∈ (Pairsproper‘𝑋) ↔ 𝑃 ∈ {𝑝 ∈ 𝒫 𝑋 ∣ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})}))
3		eqeq1 2742	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (𝑝 = {𝑎, 𝑏} ↔ 𝑃 = {𝑎, 𝑏}))
4	3	anbi2d 629	. . . . . 6 ⊢ (𝑝 = 𝑃 → ((𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏}) ↔ (𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏})))
5	4	2rexbidv 3229	. . . . 5 ⊢ (𝑝 = 𝑃 → (∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏}) ↔ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏})))
6	5	elrab 3624	. . . 4 ⊢ (𝑃 ∈ {𝑝 ∈ 𝒫 𝑋 ∣ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})} ↔ (𝑃 ∈ 𝒫 𝑋 ∧ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏})))
7	2, 6	bitrdi 287	. . 3 ⊢ (𝑋 ∈ V → (𝑃 ∈ (Pairsproper‘𝑋) ↔ (𝑃 ∈ 𝒫 𝑋 ∧ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏}))))
8		pm3.22 460	. . . . . . . . 9 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏}) → (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏))
9	8	a1i 11	. . . . . . . 8 ⊢ ((𝑃 ∈ 𝒫 𝑋 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏}) → (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)))
10	9	reximdvva 3206	. . . . . . 7 ⊢ (𝑃 ∈ 𝒫 𝑋 → (∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏}) → ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)))
11	10	imp 407	. . . . . 6 ⊢ ((𝑃 ∈ 𝒫 𝑋 ∧ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏})) → ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏))
12	11	anim2i 617	. . . . 5 ⊢ ((𝑋 ∈ V ∧ (𝑃 ∈ 𝒫 𝑋 ∧ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏}))) → (𝑋 ∈ V ∧ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)))
13	12	ex 413	. . . 4 ⊢ (𝑋 ∈ V → ((𝑃 ∈ 𝒫 𝑋 ∧ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏})) → (𝑋 ∈ V ∧ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏))))
14		simpr 485	. . . . . . . . . 10 ⊢ (((𝑋 ∈ V ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) → (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏))
15	14	ancomd 462	. . . . . . . . 9 ⊢ (((𝑋 ∈ V ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) → (𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏}))
16		prelpwi 5363	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → {𝑎, 𝑏} ∈ 𝒫 𝑋)
17	16	adantl 482	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ V ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑎, 𝑏} ∈ 𝒫 𝑋)
18	17	adantr 481	. . . . . . . . . 10 ⊢ (((𝑋 ∈ V ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) → {𝑎, 𝑏} ∈ 𝒫 𝑋)
19		eleq1 2826	. . . . . . . . . . . 12 ⊢ (𝑃 = {𝑎, 𝑏} → (𝑃 ∈ 𝒫 𝑋 ↔ {𝑎, 𝑏} ∈ 𝒫 𝑋))
20	19	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏) → (𝑃 ∈ 𝒫 𝑋 ↔ {𝑎, 𝑏} ∈ 𝒫 𝑋))
21	20	adantl 482	. . . . . . . . . 10 ⊢ (((𝑋 ∈ V ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) → (𝑃 ∈ 𝒫 𝑋 ↔ {𝑎, 𝑏} ∈ 𝒫 𝑋))
22	18, 21	mpbird 256	. . . . . . . . 9 ⊢ (((𝑋 ∈ V ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) → 𝑃 ∈ 𝒫 𝑋)
23	15, 22	jca 512	. . . . . . . 8 ⊢ (((𝑋 ∈ V ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) → ((𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏}) ∧ 𝑃 ∈ 𝒫 𝑋))
24	23	ex 413	. . . . . . 7 ⊢ ((𝑋 ∈ V ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏) → ((𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏}) ∧ 𝑃 ∈ 𝒫 𝑋)))
25	24	reximdvva 3206	. . . . . 6 ⊢ (𝑋 ∈ V → (∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏) → ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 ((𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏}) ∧ 𝑃 ∈ 𝒫 𝑋)))
26	25	imp 407	. . . . 5 ⊢ ((𝑋 ∈ V ∧ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) → ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 ((𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏}) ∧ 𝑃 ∈ 𝒫 𝑋))
27		r19.41vv 3278	. . . . . 6 ⊢ (∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 ((𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏}) ∧ 𝑃 ∈ 𝒫 𝑋) ↔ (∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏}) ∧ 𝑃 ∈ 𝒫 𝑋))
28	27	biancomi 463	. . . . 5 ⊢ (∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 ((𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏}) ∧ 𝑃 ∈ 𝒫 𝑋) ↔ (𝑃 ∈ 𝒫 𝑋 ∧ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏})))
29	26, 28	sylib 217	. . . 4 ⊢ ((𝑋 ∈ V ∧ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) → (𝑃 ∈ 𝒫 𝑋 ∧ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏})))
30	13, 29	impbid1 224	. . 3 ⊢ (𝑋 ∈ V → ((𝑃 ∈ 𝒫 𝑋 ∧ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏})) ↔ (𝑋 ∈ V ∧ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏))))
31	7, 30	bitrd 278	. 2 ⊢ (𝑋 ∈ V → (𝑃 ∈ (Pairsproper‘𝑋) ↔ (𝑋 ∈ V ∧ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏))))
32		fvprc 6766	. . . 4 ⊢ (¬ 𝑋 ∈ V → (Pairsproper‘𝑋) = ∅)
33	32	eleq2d 2824	. . 3 ⊢ (¬ 𝑋 ∈ V → (𝑃 ∈ (Pairsproper‘𝑋) ↔ 𝑃 ∈ ∅))
34		noel 4264	. . . . 5 ⊢ ¬ 𝑃 ∈ ∅
35		pm2.21 123	. . . . 5 ⊢ (¬ 𝑃 ∈ ∅ → (𝑃 ∈ ∅ → (𝑋 ∈ V ∧ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏))))
36	34, 35	mp1i 13	. . . 4 ⊢ (¬ 𝑋 ∈ V → (𝑃 ∈ ∅ → (𝑋 ∈ V ∧ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏))))
37		pm2.21 123	. . . . 5 ⊢ (¬ 𝑋 ∈ V → (𝑋 ∈ V → (∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏) → 𝑃 ∈ ∅)))
38	37	impd 411	. . . 4 ⊢ (¬ 𝑋 ∈ V → ((𝑋 ∈ V ∧ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) → 𝑃 ∈ ∅))
39	36, 38	impbid 211	. . 3 ⊢ (¬ 𝑋 ∈ V → (𝑃 ∈ ∅ ↔ (𝑋 ∈ V ∧ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏))))
40	33, 39	bitrd 278	. 2 ⊢ (¬ 𝑋 ∈ V → (𝑃 ∈ (Pairsproper‘𝑋) ↔ (𝑋 ∈ V ∧ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏))))
41	31, 40	pm2.61i 182	1 ⊢ (𝑃 ∈ (Pairsproper‘𝑋) ↔ (𝑋 ∈ V ∧ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∃wrex 3065  {crab 3068  Vcvv 3432  ∅c0 4256  𝒫 cpw 4533  {cpr 4563  ‘cfv 6433  Pairs_propercprpr 44964

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-prpr 44965

This theorem is referenced by:  inlinecirc02p  46133