Theorem disjen 9074

Description: A stronger form of pwuninel 8227. We can use pwuninel 8227, 2pwuninel 9072 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 7982	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴}) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
2	1	ad2antll 730	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴}))) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
3		simprl 771	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴}))) → 𝑥 ∈ 𝐴)
4	2, 3	eqeltrrd 2838	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴}))) → ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ 𝐴)
5		fvex 6855	. . . . . . 7 ⊢ (1st ‘𝑥) ∈ V
6		fvex 6855	. . . . . . 7 ⊢ (2nd ‘𝑥) ∈ V
7	5, 6	opelrn 5900	. . . . . 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 8227	. . . . . 6 ⊢ ¬ 𝒫 ∪ ran 𝐴 ∈ ran 𝐴
10		xp2nd 7976	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴}) → (2nd ‘𝑥) ∈ {𝒫 ∪ ran 𝐴})
11	10	ad2antll 730	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴}))) → (2nd ‘𝑥) ∈ {𝒫 ∪ ran 𝐴})
12		elsni 4599	. . . . . . . 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 2822	. . . . . 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 817	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴})))
17		elin 3919	. . . 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 4361	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∩ (𝐵 × {𝒫 ∪ ran 𝐴})) = ∅)
20		simpr 484	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ 𝑊)
21		rnexg 7854	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V)
22	21	adantr 480	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ran 𝐴 ∈ V)
23		uniexg 7695	. . . 4 ⊢ (ran 𝐴 ∈ V → ∪ ran 𝐴 ∈ V)
24		pwexg 5325	. . . 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 9002	. . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝒫 ∪ ran 𝐴 ∈ V) → (𝐵 × {𝒫 ∪ ran 𝐴}) ≈ 𝐵)
27	20, 25, 26	syl2anc 585	. 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 1542   ∈ wcel 2114  Vcvv 3442   ∩ cin 3902  ∅c0 4287  𝒫 cpw 4556  {csn 4582  ⟨cop 4588  ∪ cuni 4865   class class class wbr 5100   × cxp 5630  ran crn 5633  ‘cfv 6500  1^st c1st 7941  2^nd c2nd 7942   ≈ cen 8892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-1st 7943  df-2nd 7944  df-en 8896

This theorem is referenced by:  disjenex  9075  domss2  9076