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 ∈ (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 44640	. . . . 5 ⊢ (𝑋 ∈ V → (Pairsproper‘𝑋) = {𝑝 ∈ 𝒫 𝑋 ∣ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})})
2	1	eleq2d 2823	. . . 4 ⊢ (𝑋 ∈ V → (𝑃 ∈ (Pairsproper‘𝑋) ↔ 𝑃 ∈ {𝑝 ∈ 𝒫 𝑋 ∣ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})}))
3		eqeq1 2741	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (𝑝 = {𝑎, 𝑏} ↔ 𝑃 = {𝑎, 𝑏}))
4	3	anbi2d 632	. . . . . 6 ⊢ (𝑝 = 𝑃 → ((𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏}) ↔ (𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏})))
5	4	2rexbidv 3219	. . . . 5 ⊢ (𝑝 = 𝑃 → (∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏}) ↔ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏})))
6	5	elrab 3602	. . . 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 290	. . 3 ⊢ (𝑋 ∈ V → (𝑃 ∈ (Pairsproper‘𝑋) ↔ (𝑃 ∈ 𝒫 𝑋 ∧ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏}))))
8		pm3.22 463	. . . . . . . . 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 3196	. . . . . . 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 410	. . . . . 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 620	. . . . 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 416	. . . 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 488	. . . . . . . . . 10 ⊢ (((𝑋 ∈ V ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) → (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏))
15	14	ancomd 465	. . . . . . . . 9 ⊢ (((𝑋 ∈ V ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) → (𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏}))
16		prelpwi 5332	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → {𝑎, 𝑏} ∈ 𝒫 𝑋)
17	16	adantl 485	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ V ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑎, 𝑏} ∈ 𝒫 𝑋)
18	17	adantr 484	. . . . . . . . . 10 ⊢ (((𝑋 ∈ V ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) → {𝑎, 𝑏} ∈ 𝒫 𝑋)
19		eleq1 2825	. . . . . . . . . . . 12 ⊢ (𝑃 = {𝑎, 𝑏} → (𝑃 ∈ 𝒫 𝑋 ↔ {𝑎, 𝑏} ∈ 𝒫 𝑋))
20	19	adantr 484	. . . . . . . . . . 11 ⊢ ((𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏) → (𝑃 ∈ 𝒫 𝑋 ↔ {𝑎, 𝑏} ∈ 𝒫 𝑋))
21	20	adantl 485	. . . . . . . . . 10 ⊢ (((𝑋 ∈ V ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) → (𝑃 ∈ 𝒫 𝑋 ↔ {𝑎, 𝑏} ∈ 𝒫 𝑋))
22	18, 21	mpbird 260	. . . . . . . . 9 ⊢ (((𝑋 ∈ V ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) → 𝑃 ∈ 𝒫 𝑋)
23	15, 22	jca 515	. . . . . . . 8 ⊢ (((𝑋 ∈ V ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) → ((𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏}) ∧ 𝑃 ∈ 𝒫 𝑋))
24	23	ex 416	. . . . . . 7 ⊢ ((𝑋 ∈ V ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏) → ((𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏}) ∧ 𝑃 ∈ 𝒫 𝑋)))
25	24	reximdvva 3196	. . . . . 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 410	. . . . 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 3262	. . . . . 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 466	. . . . 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 221	. . . 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 228	. . 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 282	. 2 ⊢ (𝑋 ∈ V → (𝑃 ∈ (Pairsproper‘𝑋) ↔ (𝑋 ∈ V ∧ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏))))
32		fvprc 6709	. . . 4 ⊢ (¬ 𝑋 ∈ V → (Pairsproper‘𝑋) = ∅)
33	32	eleq2d 2823	. . 3 ⊢ (¬ 𝑋 ∈ V → (𝑃 ∈ (Pairsproper‘𝑋) ↔ 𝑃 ∈ ∅))
34		noel 4245	. . . . 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 414	. . . 4 ⊢ (¬ 𝑋 ∈ V → ((𝑋 ∈ V ∧ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) → 𝑃 ∈ ∅))
39	36, 38	impbid 215	. . 3 ⊢ (¬ 𝑋 ∈ V → (𝑃 ∈ ∅ ↔ (𝑋 ∈ V ∧ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏))))
40	33, 39	bitrd 282	. 2 ⊢ (¬ 𝑋 ∈ V → (𝑃 ∈ (Pairsproper‘𝑋) ↔ (𝑋 ∈ V ∧ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏))))
41	31, 40	pm2.61i 185	1 ⊢ (𝑃 ∈ (Pairsproper‘𝑋) ↔ (𝑋 ∈ V ∧ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)))

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  Pairs_propercprpr 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