Theorem prprelprb 44857

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 44855	. . . . 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 628	. . . . . 6 ⊢ (𝑝 = 𝑃 → ((𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏}) ↔ (𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏})))
5	4	2rexbidv 3228	. . . . 5 ⊢ (𝑝 = 𝑃 → (∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏}) ↔ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏})))
6	5	elrab 3617	. . . 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 286	. . 3 ⊢ (𝑋 ∈ V → (𝑃 ∈ (Pairsproper‘𝑋) ↔ (𝑃 ∈ 𝒫 𝑋 ∧ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏}))))
8		pm3.22 459	. . . . . . . . 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 3205	. . . . . . 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 406	. . . . . 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 616	. . . . 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 412	. . . 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 484	. . . . . . . . . 10 ⊢ (((𝑋 ∈ V ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) → (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏))
15	14	ancomd 461	. . . . . . . . 9 ⊢ (((𝑋 ∈ V ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) → (𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏}))
16		prelpwi 5357	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → {𝑎, 𝑏} ∈ 𝒫 𝑋)
17	16	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ V ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑎, 𝑏} ∈ 𝒫 𝑋)
18	17	adantr 480	. . . . . . . . . 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 480	. . . . . . . . . . 11 ⊢ ((𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏) → (𝑃 ∈ 𝒫 𝑋 ↔ {𝑎, 𝑏} ∈ 𝒫 𝑋))
21	20	adantl 481	. . . . . . . . . 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 511	. . . . . . . 8 ⊢ (((𝑋 ∈ V ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) → ((𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏}) ∧ 𝑃 ∈ 𝒫 𝑋))
24	23	ex 412	. . . . . . 7 ⊢ ((𝑋 ∈ V ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏) → ((𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏}) ∧ 𝑃 ∈ 𝒫 𝑋)))
25	24	reximdvva 3205	. . . . . 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 406	. . . . 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 3275	. . . . . 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 462	. . . . 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 6748	. . . 4 ⊢ (¬ 𝑋 ∈ V → (Pairsproper‘𝑋) = ∅)
33	32	eleq2d 2824	. . 3 ⊢ (¬ 𝑋 ∈ V → (𝑃 ∈ (Pairsproper‘𝑋) ↔ 𝑃 ∈ ∅))
34		noel 4261	. . . . 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 410	. . . 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 395   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∃wrex 3064  {crab 3067  Vcvv 3422  ∅c0 4253  𝒫 cpw 4530  {cpr 4560  ‘cfv 6418  Pairs_propercprpr 44852

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-prpr 44853

This theorem is referenced by:  inlinecirc02p  46021