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Theorem bj-2uplex 35600
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 35591 . . . 4 pr1𝐴, 𝐵⦆ = 𝐴
2 bj-pr1ex 35584 . . . 4 (⦅𝐴, 𝐵⦆ ∈ V → pr1𝐴, 𝐵⦆ ∈ V)
31, 2eqeltrrid 2837 . . 3 (⦅𝐴, 𝐵⦆ ∈ V → 𝐴 ∈ V)
4 bj-pr22val 35597 . . . 4 pr2𝐴, 𝐵⦆ = 𝐵
5 bj-pr2ex 35598 . . . 4 (⦅𝐴, 𝐵⦆ ∈ V → pr2𝐴, 𝐵⦆ ∈ V)
64, 5eqeltrrid 2837 . . 3 (⦅𝐴, 𝐵⦆ ∈ V → 𝐵 ∈ V)
73, 6jca 512 . 2 (⦅𝐴, 𝐵⦆ ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V))
8 df-bj-2upl 35589 . . 3 𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
9 bj-1uplex 35586 . . . . 5 (⦅𝐴⦆ ∈ V ↔ 𝐴 ∈ V)
109biimpri 227 . . . 4 (𝐴 ∈ V → ⦅𝐴⦆ ∈ V)
11 snex 5408 . . . . 5 {1o} ∈ V
12 bj-xtagex 35567 . . . . 5 ({1o} ∈ V → (𝐵 ∈ V → ({1o} × tag 𝐵) ∈ V))
1311, 12ax-mp 5 . . . 4 (𝐵 ∈ V → ({1o} × tag 𝐵) ∈ V)
14 unexg 7703 . . . 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 2836 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⦅𝐴, 𝐵⦆ ∈ V)
177, 16impbii 208 1 (⦅𝐴, 𝐵⦆ ∈ V ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wcel 2106  Vcvv 3459  cun 3926  {csn 4606   × cxp 5651  1oc1o 8425  tag bj-ctag 35552  bj-c1upl 35575  pr1 bj-cpr1 35578  bj-c2uple 35588  pr2 bj-cpr2 35592
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5262  ax-sep 5276  ax-nul 5283  ax-pow 5340  ax-pr 5404  ax-un 7692
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-if 4507  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-br 5126  df-opab 5188  df-xp 5659  df-rel 5660  df-cnv 5661  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-suc 6343  df-1o 8432  df-bj-sngl 35544  df-bj-tag 35553  df-bj-proj 35569  df-bj-1upl 35576  df-bj-pr1 35579  df-bj-2upl 35589  df-bj-pr2 35593
This theorem is referenced by: (None)
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