Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-2uplex Structured version   Visualization version   GIF version

Theorem bj-2uplex 37023
Description: A couple is a set if and only if its coordinates are sets. For the advantages offered by the reverse closure property, see the section head comment. (Contributed by BJ, 6-Oct-2018.)
Assertion
Ref Expression
bj-2uplex (⦅𝐴, 𝐵⦆ ∈ V ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))

Proof of Theorem bj-2uplex
StepHypRef Expression
1 bj-pr21val 37014 . . . 4 pr1𝐴, 𝐵⦆ = 𝐴
2 bj-pr1ex 37007 . . . 4 (⦅𝐴, 𝐵⦆ ∈ V → pr1𝐴, 𝐵⦆ ∈ V)
31, 2eqeltrrid 2846 . . 3 (⦅𝐴, 𝐵⦆ ∈ V → 𝐴 ∈ V)
4 bj-pr22val 37020 . . . 4 pr2𝐴, 𝐵⦆ = 𝐵
5 bj-pr2ex 37021 . . . 4 (⦅𝐴, 𝐵⦆ ∈ V → pr2𝐴, 𝐵⦆ ∈ V)
64, 5eqeltrrid 2846 . . 3 (⦅𝐴, 𝐵⦆ ∈ V → 𝐵 ∈ V)
73, 6jca 511 . 2 (⦅𝐴, 𝐵⦆ ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V))
8 df-bj-2upl 37012 . . 3 𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
9 bj-1uplex 37009 . . . . 5 (⦅𝐴⦆ ∈ V ↔ 𝐴 ∈ V)
109biimpri 228 . . . 4 (𝐴 ∈ V → ⦅𝐴⦆ ∈ V)
11 snex 5436 . . . . 5 {1o} ∈ V
12 bj-xtagex 36990 . . . . 5 ({1o} ∈ V → (𝐵 ∈ V → ({1o} × tag 𝐵) ∈ V))
1311, 12ax-mp 5 . . . 4 (𝐵 ∈ V → ({1o} × tag 𝐵) ∈ V)
14 unexg 7763 . . . 4 ((⦅𝐴⦆ ∈ V ∧ ({1o} × tag 𝐵) ∈ V) → (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) ∈ V)
1510, 13, 14syl2an 596 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) ∈ V)
168, 15eqeltrid 2845 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⦅𝐴, 𝐵⦆ ∈ V)
177, 16impbii 209 1 (⦅𝐴, 𝐵⦆ ∈ V ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2108  Vcvv 3480  cun 3949  {csn 4626   × cxp 5683  1oc1o 8499  tag bj-ctag 36975  bj-c1upl 36998  pr1 bj-cpr1 37001  bj-c2uple 37011  pr2 bj-cpr2 37015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-suc 6390  df-1o 8506  df-bj-sngl 36967  df-bj-tag 36976  df-bj-proj 36992  df-bj-1upl 36999  df-bj-pr1 37002  df-bj-2upl 37012  df-bj-pr2 37016
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator