MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  disjen Structured version   Visualization version   GIF version

Theorem disjen 9133
Description: A stronger form of pwuninel 8258. We can use pwuninel 8258, 2pwuninel 9131 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 8010 . . . . . . . 8 (𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
21ad2antll 726 . . . . . . 7 (((𝐴𝑉𝐵𝑊) ∧ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}))) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
3 simprl 768 . . . . . . 7 (((𝐴𝑉𝐵𝑊) ∧ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}))) → 𝑥𝐴)
42, 3eqeltrrd 2828 . . . . . 6 (((𝐴𝑉𝐵𝑊) ∧ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}))) → ⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝐴)
5 fvex 6897 . . . . . . 7 (1st𝑥) ∈ V
6 fvex 6897 . . . . . . 7 (2nd𝑥) ∈ V
75, 6opelrn 5935 . . . . . 6 (⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝐴 → (2nd𝑥) ∈ ran 𝐴)
84, 7syl 17 . . . . 5 (((𝐴𝑉𝐵𝑊) ∧ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}))) → (2nd𝑥) ∈ ran 𝐴)
9 pwuninel 8258 . . . . . 6 ¬ 𝒫 ran 𝐴 ∈ ran 𝐴
10 xp2nd 8004 . . . . . . . . 9 (𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}) → (2nd𝑥) ∈ {𝒫 ran 𝐴})
1110ad2antll 726 . . . . . . . 8 (((𝐴𝑉𝐵𝑊) ∧ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}))) → (2nd𝑥) ∈ {𝒫 ran 𝐴})
12 elsni 4640 . . . . . . . 8 ((2nd𝑥) ∈ {𝒫 ran 𝐴} → (2nd𝑥) = 𝒫 ran 𝐴)
1311, 12syl 17 . . . . . . 7 (((𝐴𝑉𝐵𝑊) ∧ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}))) → (2nd𝑥) = 𝒫 ran 𝐴)
1413eleq1d 2812 . . . . . 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 814 . . . 4 ((𝐴𝑉𝐵𝑊) → ¬ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴})))
17 elin 3959 . . . 4 (𝑥 ∈ (𝐴 ∩ (𝐵 × {𝒫 ran 𝐴})) ↔ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴})))
1816, 17sylnibr 329 . . 3 ((𝐴𝑉𝐵𝑊) → ¬ 𝑥 ∈ (𝐴 ∩ (𝐵 × {𝒫 ran 𝐴})))
1918eq0rdv 4399 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴 ∩ (𝐵 × {𝒫 ran 𝐴})) = ∅)
20 simpr 484 . . 3 ((𝐴𝑉𝐵𝑊) → 𝐵𝑊)
21 rnexg 7891 . . . . 5 (𝐴𝑉 → ran 𝐴 ∈ V)
2221adantr 480 . . . 4 ((𝐴𝑉𝐵𝑊) → ran 𝐴 ∈ V)
23 uniexg 7726 . . . 4 (ran 𝐴 ∈ V → ran 𝐴 ∈ V)
24 pwexg 5369 . . . 4 ( ran 𝐴 ∈ V → 𝒫 ran 𝐴 ∈ V)
2522, 23, 243syl 18 . . 3 ((𝐴𝑉𝐵𝑊) → 𝒫 ran 𝐴 ∈ V)
26 xpsneng 9055 . . 3 ((𝐵𝑊 ∧ 𝒫 ran 𝐴 ∈ V) → (𝐵 × {𝒫 ran 𝐴}) ≈ 𝐵)
2720, 25, 26syl2anc 583 . 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 1533  wcel 2098  Vcvv 3468  cin 3942  c0 4317  𝒫 cpw 4597  {csn 4623  cop 4629   cuni 4902   class class class wbr 5141   × cxp 5667  ran crn 5670  cfv 6536  1st c1st 7969  2nd c2nd 7970  cen 8935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-1st 7971  df-2nd 7972  df-en 8939
This theorem is referenced by:  disjenex  9134  domss2  9135
  Copyright terms: Public domain W3C validator