Theorem disjen 9060

Description: A stronger form of pwuninel 8215. We can use pwuninel 8215, 2pwuninel 9058 to create one or two sets disjoint from a given set 𝐴, but here we show that in fact such constructions exist for arbitrarily large disjoint extensions, which is to say that for any set 𝐵 we can construct a set 𝑥 that is equinumerous to it and disjoint from 𝐴. (Contributed by Mario Carneiro, 7-Feb-2015.)

Assertion

Ref	Expression
disjen	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∩ (𝐵 × {𝒫 ∪ ran 𝐴})) = ∅ ∧ (𝐵 × {𝒫 ∪ ran 𝐴}) ≈ 𝐵))

Proof of Theorem disjen

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1st2nd2 7970	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴}) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
2	1	ad2antll 729	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴}))) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
3		simprl 770	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴}))) → 𝑥 ∈ 𝐴)
4	2, 3	eqeltrrd 2835	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴}))) → ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ 𝐴)
5		fvex 6845	. . . . . . 7 ⊢ (1st ‘𝑥) ∈ V
6		fvex 6845	. . . . . . 7 ⊢ (2nd ‘𝑥) ∈ V
7	5, 6	opelrn 5890	. . . . . 6 ⊢ (⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ 𝐴 → (2nd ‘𝑥) ∈ ran 𝐴)
8	4, 7	syl 17	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴}))) → (2nd ‘𝑥) ∈ ran 𝐴)
9		pwuninel 8215	. . . . . 6 ⊢ ¬ 𝒫 ∪ ran 𝐴 ∈ ran 𝐴
10		xp2nd 7964	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴}) → (2nd ‘𝑥) ∈ {𝒫 ∪ ran 𝐴})
11	10	ad2antll 729	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴}))) → (2nd ‘𝑥) ∈ {𝒫 ∪ ran 𝐴})
12		elsni 4595	. . . . . . . 8 ⊢ ((2nd ‘𝑥) ∈ {𝒫 ∪ ran 𝐴} → (2nd ‘𝑥) = 𝒫 ∪ ran 𝐴)
13	11, 12	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴}))) → (2nd ‘𝑥) = 𝒫 ∪ ran 𝐴)
14	13	eleq1d 2819	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴}))) → ((2nd ‘𝑥) ∈ ran 𝐴 ↔ 𝒫 ∪ ran 𝐴 ∈ ran 𝐴))
15	9, 14	mtbiri 327	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴}))) → ¬ (2nd ‘𝑥) ∈ ran 𝐴)
16	8, 15	pm2.65da 816	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴})))
17		elin 3915	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∩ (𝐵 × {𝒫 ∪ ran 𝐴})) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴})))
18	16, 17	sylnibr 329	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ¬ 𝑥 ∈ (𝐴 ∩ (𝐵 × {𝒫 ∪ ran 𝐴})))
19	18	eq0rdv 4357	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∩ (𝐵 × {𝒫 ∪ ran 𝐴})) = ∅)
20		simpr 484	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ 𝑊)
21		rnexg 7842	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V)
22	21	adantr 480	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ran 𝐴 ∈ V)
23		uniexg 7683	. . . 4 ⊢ (ran 𝐴 ∈ V → ∪ ran 𝐴 ∈ V)
24		pwexg 5321	. . . 4 ⊢ (∪ ran 𝐴 ∈ V → 𝒫 ∪ ran 𝐴 ∈ V)
25	22, 23, 24	3syl 18	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 ∪ ran 𝐴 ∈ V)
26		xpsneng 8988	. . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝒫 ∪ ran 𝐴 ∈ V) → (𝐵 × {𝒫 ∪ ran 𝐴}) ≈ 𝐵)
27	20, 25, 26	syl2anc 584	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 × {𝒫 ∪ ran 𝐴}) ≈ 𝐵)
28	19, 27	jca 511	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∩ (𝐵 × {𝒫 ∪ ran 𝐴})) = ∅ ∧ (𝐵 × {𝒫 ∪ ran 𝐴}) ≈ 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3438   ∩ cin 3898  ∅c0 4283  𝒫 cpw 4552  {csn 4578  ⟨cop 4584  ∪ cuni 4861   class class class wbr 5096   × cxp 5620  ran crn 5623  ‘cfv 6490  1^st c1st 7929  2^nd c2nd 7930   ≈ cen 8878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-1st 7931  df-2nd 7932  df-en 8882

This theorem is referenced by:  disjenex  9061  domss2  9062