Theorem prprelprb 43072

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 43070	. . . . 5 ⊢ (𝑋 ∈ V → (Pairsproper‘𝑋) = {𝑝 ∈ 𝒫 𝑋 ∣ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})})
2	1	eleq2d 2845	. . . 4 ⊢ (𝑋 ∈ V → (𝑃 ∈ (Pairsproper‘𝑋) ↔ 𝑃 ∈ {𝑝 ∈ 𝒫 𝑋 ∣ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})}))
3		eqeq1 2776	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (𝑝 = {𝑎, 𝑏} ↔ 𝑃 = {𝑎, 𝑏}))
4	3	anbi2d 619	. . . . . 6 ⊢ (𝑝 = 𝑃 → ((𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏}) ↔ (𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏})))
5	4	2rexbidv 3239	. . . . 5 ⊢ (𝑝 = 𝑃 → (∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏}) ↔ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏})))
6	5	elrab 3589	. . . 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	syl6bb 279	. . 3 ⊢ (𝑋 ∈ V → (𝑃 ∈ (Pairsproper‘𝑋) ↔ (𝑃 ∈ 𝒫 𝑋 ∧ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏}))))
8		pm3.22 452	. . . . . . . . 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 3216	. . . . . . 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 398	. . . . . 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 607	. . . . 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 405	. . . 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 477	. . . . . . . . . 10 ⊢ (((𝑋 ∈ V ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) → (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏))
15	14	ancomd 454	. . . . . . . . 9 ⊢ (((𝑋 ∈ V ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) → (𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏}))
16		prelpwi 5192	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → {𝑎, 𝑏} ∈ 𝒫 𝑋)
17	16	adantl 474	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ V ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑎, 𝑏} ∈ 𝒫 𝑋)
18	17	adantr 473	. . . . . . . . . 10 ⊢ (((𝑋 ∈ V ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) → {𝑎, 𝑏} ∈ 𝒫 𝑋)
19		eleq1 2847	. . . . . . . . . . . 12 ⊢ (𝑃 = {𝑎, 𝑏} → (𝑃 ∈ 𝒫 𝑋 ↔ {𝑎, 𝑏} ∈ 𝒫 𝑋))
20	19	adantr 473	. . . . . . . . . . 11 ⊢ ((𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏) → (𝑃 ∈ 𝒫 𝑋 ↔ {𝑎, 𝑏} ∈ 𝒫 𝑋))
21	20	adantl 474	. . . . . . . . . 10 ⊢ (((𝑋 ∈ V ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) → (𝑃 ∈ 𝒫 𝑋 ↔ {𝑎, 𝑏} ∈ 𝒫 𝑋))
22	18, 21	mpbird 249	. . . . . . . . 9 ⊢ (((𝑋 ∈ V ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) → 𝑃 ∈ 𝒫 𝑋)
23	15, 22	jca 504	. . . . . . . 8 ⊢ (((𝑋 ∈ V ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) → ((𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏}) ∧ 𝑃 ∈ 𝒫 𝑋))
24	23	ex 405	. . . . . . 7 ⊢ ((𝑋 ∈ V ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏) → ((𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏}) ∧ 𝑃 ∈ 𝒫 𝑋)))
25	24	reximdvva 3216	. . . . . 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 398	. . . . 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 3284	. . . . . 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 455	. . . . 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 210	. . . 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 217	. . 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 271	. 2 ⊢ (𝑋 ∈ V → (𝑃 ∈ (Pairsproper‘𝑋) ↔ (𝑋 ∈ V ∧ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏))))
32		fvprc 6489	. . . 4 ⊢ (¬ 𝑋 ∈ V → (Pairsproper‘𝑋) = ∅)
33	32	eleq2d 2845	. . 3 ⊢ (¬ 𝑋 ∈ V → (𝑃 ∈ (Pairsproper‘𝑋) ↔ 𝑃 ∈ ∅))
34		noel 4177	. . . . 5 ⊢ ¬ 𝑃 ∈ ∅
35		pm2.21 121	. . . . 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		pm2.21 121	. . . . 5 ⊢ (¬ 𝑋 ∈ V → (𝑋 ∈ V → (∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏) → 𝑃 ∈ ∅)))
38	37	impd 402	. . . 4 ⊢ (¬ 𝑋 ∈ V → ((𝑋 ∈ V ∧ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) → 𝑃 ∈ ∅))
39	36, 38	impbid 204	. . 3 ⊢ (¬ 𝑋 ∈ V → (𝑃 ∈ ∅ ↔ (𝑋 ∈ V ∧ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏))))
40	33, 39	bitrd 271	. 2 ⊢ (¬ 𝑋 ∈ V → (𝑃 ∈ (Pairsproper‘𝑋) ↔ (𝑋 ∈ V ∧ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏))))
41	31, 40	pm2.61i 177	1 ⊢ (𝑃 ∈ (Pairsproper‘𝑋) ↔ (𝑋 ∈ V ∧ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1507   ∈ wcel 2050   ≠ wne 2961  ∃wrex 3083  {crab 3086  Vcvv 3409  ∅c0 4172  𝒫 cpw 4416  {cpr 4437  ‘cfv 6185  Pairs_propercprpr 43067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-prpr 43068

This theorem is referenced by:  inlinecirc02p  44167