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 34231
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 34222 . . . 4 pr1𝐴, 𝐵⦆ = 𝐴
2 bj-pr1ex 34215 . . . 4 (⦅𝐴, 𝐵⦆ ∈ V → pr1𝐴, 𝐵⦆ ∈ V)
31, 2eqeltrrid 2915 . . 3 (⦅𝐴, 𝐵⦆ ∈ V → 𝐴 ∈ V)
4 bj-pr22val 34228 . . . 4 pr2𝐴, 𝐵⦆ = 𝐵
5 bj-pr2ex 34229 . . . 4 (⦅𝐴, 𝐵⦆ ∈ V → pr2𝐴, 𝐵⦆ ∈ V)
64, 5eqeltrrid 2915 . . 3 (⦅𝐴, 𝐵⦆ ∈ V → 𝐵 ∈ V)
73, 6jca 512 . 2 (⦅𝐴, 𝐵⦆ ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V))
8 df-bj-2upl 34220 . . 3 𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
9 bj-1uplex 34217 . . . . 5 (⦅𝐴⦆ ∈ V ↔ 𝐴 ∈ V)
109biimpri 229 . . . 4 (𝐴 ∈ V → ⦅𝐴⦆ ∈ V)
11 snex 5322 . . . . 5 {1o} ∈ V
12 bj-xtagex 34198 . . . . 5 ({1o} ∈ V → (𝐵 ∈ V → ({1o} × tag 𝐵) ∈ V))
1311, 12ax-mp 5 . . . 4 (𝐵 ∈ V → ({1o} × tag 𝐵) ∈ V)
14 unexg 7461 . . . 4 ((⦅𝐴⦆ ∈ V ∧ ({1o} × tag 𝐵) ∈ V) → (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) ∈ V)
1510, 13, 14syl2an 595 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) ∈ V)
168, 15eqeltrid 2914 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⦅𝐴, 𝐵⦆ ∈ V)
177, 16impbii 210 1 (⦅𝐴, 𝐵⦆ ∈ V ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wcel 2105  Vcvv 3492  cun 3931  {csn 4557   × cxp 5546  1oc1o 8084  tag bj-ctag 34183  bj-c1upl 34206  pr1 bj-cpr1 34209  bj-c2uple 34219  pr2 bj-cpr2 34223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-tr 5164  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-suc 6190  df-1o 8091  df-bj-sngl 34175  df-bj-tag 34184  df-bj-proj 34200  df-bj-1upl 34207  df-bj-pr1 34210  df-bj-2upl 34220  df-bj-pr2 34224
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator