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Theorem disjen 9175
Description: A stronger form of pwuninel 8301. We can use pwuninel 8301, 2pwuninel 9173 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 ((𝐴𝑉𝐵𝑊) → ((𝐴 ∩ (𝐵 × {𝒫 ran 𝐴})) = ∅ ∧ (𝐵 × {𝒫 ran 𝐴}) ≈ 𝐵))

Proof of Theorem disjen
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 1st2nd2 8054 . . . . . . . 8 (𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
21ad2antll 729 . . . . . . 7 (((𝐴𝑉𝐵𝑊) ∧ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}))) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
3 simprl 770 . . . . . . 7 (((𝐴𝑉𝐵𝑊) ∧ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}))) → 𝑥𝐴)
42, 3eqeltrrd 2841 . . . . . 6 (((𝐴𝑉𝐵𝑊) ∧ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}))) → ⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝐴)
5 fvex 6918 . . . . . . 7 (1st𝑥) ∈ V
6 fvex 6918 . . . . . . 7 (2nd𝑥) ∈ V
75, 6opelrn 5953 . . . . . 6 (⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝐴 → (2nd𝑥) ∈ ran 𝐴)
84, 7syl 17 . . . . 5 (((𝐴𝑉𝐵𝑊) ∧ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}))) → (2nd𝑥) ∈ ran 𝐴)
9 pwuninel 8301 . . . . . 6 ¬ 𝒫 ran 𝐴 ∈ ran 𝐴
10 xp2nd 8048 . . . . . . . . 9 (𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}) → (2nd𝑥) ∈ {𝒫 ran 𝐴})
1110ad2antll 729 . . . . . . . 8 (((𝐴𝑉𝐵𝑊) ∧ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}))) → (2nd𝑥) ∈ {𝒫 ran 𝐴})
12 elsni 4642 . . . . . . . 8 ((2nd𝑥) ∈ {𝒫 ran 𝐴} → (2nd𝑥) = 𝒫 ran 𝐴)
1311, 12syl 17 . . . . . . 7 (((𝐴𝑉𝐵𝑊) ∧ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}))) → (2nd𝑥) = 𝒫 ran 𝐴)
1413eleq1d 2825 . . . . . 6 (((𝐴𝑉𝐵𝑊) ∧ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}))) → ((2nd𝑥) ∈ ran 𝐴 ↔ 𝒫 ran 𝐴 ∈ ran 𝐴))
159, 14mtbiri 327 . . . . 5 (((𝐴𝑉𝐵𝑊) ∧ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}))) → ¬ (2nd𝑥) ∈ ran 𝐴)
168, 15pm2.65da 816 . . . 4 ((𝐴𝑉𝐵𝑊) → ¬ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴})))
17 elin 3966 . . . 4 (𝑥 ∈ (𝐴 ∩ (𝐵 × {𝒫 ran 𝐴})) ↔ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴})))
1816, 17sylnibr 329 . . 3 ((𝐴𝑉𝐵𝑊) → ¬ 𝑥 ∈ (𝐴 ∩ (𝐵 × {𝒫 ran 𝐴})))
1918eq0rdv 4406 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴 ∩ (𝐵 × {𝒫 ran 𝐴})) = ∅)
20 simpr 484 . . 3 ((𝐴𝑉𝐵𝑊) → 𝐵𝑊)
21 rnexg 7925 . . . . 5 (𝐴𝑉 → ran 𝐴 ∈ V)
2221adantr 480 . . . 4 ((𝐴𝑉𝐵𝑊) → ran 𝐴 ∈ V)
23 uniexg 7761 . . . 4 (ran 𝐴 ∈ V → ran 𝐴 ∈ V)
24 pwexg 5377 . . . 4 ( ran 𝐴 ∈ V → 𝒫 ran 𝐴 ∈ V)
2522, 23, 243syl 18 . . 3 ((𝐴𝑉𝐵𝑊) → 𝒫 ran 𝐴 ∈ V)
26 xpsneng 9097 . . 3 ((𝐵𝑊 ∧ 𝒫 ran 𝐴 ∈ V) → (𝐵 × {𝒫 ran 𝐴}) ≈ 𝐵)
2720, 25, 26syl2anc 584 . 2 ((𝐴𝑉𝐵𝑊) → (𝐵 × {𝒫 ran 𝐴}) ≈ 𝐵)
2819, 27jca 511 1 ((𝐴𝑉𝐵𝑊) → ((𝐴 ∩ (𝐵 × {𝒫 ran 𝐴})) = ∅ ∧ (𝐵 × {𝒫 ran 𝐴}) ≈ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  Vcvv 3479  cin 3949  c0 4332  𝒫 cpw 4599  {csn 4625  cop 4631   cuni 4906   class class class wbr 5142   × cxp 5682  ran crn 5685  cfv 6560  1st c1st 8013  2nd c2nd 8014  cen 8983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-1st 8015  df-2nd 8016  df-en 8987
This theorem is referenced by:  disjenex  9176  domss2  9177
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