Theorem prprelprb 48543

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 48541	. . . . 5 ⊢ (𝑋 ∈ V → (Pairsproper‘𝑋) = {𝑝 ∈ 𝒫 𝑋 ∣ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})})
2	1	eleq2d 2847	. . . 4 ⊢ (𝑋 ∈ V → (𝑃 ∈ (Pairsproper‘𝑋) ↔ 𝑃 ∈ {𝑝 ∈ 𝒫 𝑋 ∣ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})}))
3		eqeq1 2765	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (𝑝 = {𝑎, 𝑏} ↔ 𝑃 = {𝑎, 𝑏}))
4	3	anbi2d 642	. . . . . 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 3645	. . . 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		pm3.22 465	. . . . . . . . 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 3211	. . . . . . 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 412	. . . . . 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 629	. . . . 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 418	. . . 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 490	. . . . . . . . . 10 ⊢ (((𝑋 ∈ V ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) → (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏))
15	14	ancomd 467	. . . . . . . . 9 ⊢ (((𝑋 ∈ V ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) → (𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏}))
16		prelpwi 5415	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → {𝑎, 𝑏} ∈ 𝒫 𝑋)
17	16	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ V ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑎, 𝑏} ∈ 𝒫 𝑋)
18	17	adantr 486	. . . . . . . . . 10 ⊢ (((𝑋 ∈ V ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) → {𝑎, 𝑏} ∈ 𝒫 𝑋)
19		eleq1 2849	. . . . . . . . . . . 12 ⊢ (𝑃 = {𝑎, 𝑏} → (𝑃 ∈ 𝒫 𝑋 ↔ {𝑎, 𝑏} ∈ 𝒫 𝑋))
20	19	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏) → (𝑃 ∈ 𝒫 𝑋 ↔ {𝑎, 𝑏} ∈ 𝒫 𝑋))
21	20	adantl 487	. . . . . . . . . 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 521	. . . . . . . 8 ⊢ (((𝑋 ∈ V ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) → ((𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏}) ∧ 𝑃 ∈ 𝒫 𝑋))
24	23	ex 418	. . . . . . 7 ⊢ ((𝑋 ∈ V ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏) → ((𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏}) ∧ 𝑃 ∈ 𝒫 𝑋)))
25	24	reximdvva 3211	. . . . . 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 412	. . . . 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 3233	. . . . . 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 468	. . . . 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 6869	. . . 4 ⊢ (¬ 𝑋 ∈ V → (Pairsproper‘𝑋) = ∅)
33	32	eleq2d 2847	. . 3 ⊢ (¬ 𝑋 ∈ V → (𝑃 ∈ (Pairsproper‘𝑋) ↔ 𝑃 ∈ ∅))
34		noel 4284	. . . . 5 ⊢ ¬ 𝑃 ∈ ∅
35		pm2.21 124	. . . . 5 ⊢ (¬ 𝑃 ∈ ∅ → (𝑃 ∈ ∅ → (𝑋 ∈ V ∧ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏))))
36	34, 35	mp1i 14	. . . 4 ⊢ (¬ 𝑋 ∈ V → (𝑃 ∈ ∅ → (𝑋 ∈ V ∧ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏))))
37		pm2.21 124	. . . . 5 ⊢ (¬ 𝑋 ∈ V → (𝑋 ∈ V → (∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏) → 𝑃 ∈ ∅)))
38	37	impd 416	. . . 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 184	1 ⊢ (𝑃 ∈ (Pairsproper‘𝑋) ↔ (𝑋 ∈ V ∧ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∃wrex 3087  {crab 3413  Vcvv 3451  ∅c0 4279  𝒫 cpw 4557  {cpr 4586  ‘cfv 6531  Pairs_propercprpr 48538

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6487  df-fun 6533  df-fv 6539  df-prpr 48539

This theorem is used by:  inlinecirc02p  49843