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Theorem bj-2uplex 37005
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 36996 . . . 4 pr1𝐴, 𝐵⦆ = 𝐴
2 bj-pr1ex 36989 . . . 4 (⦅𝐴, 𝐵⦆ ∈ V → pr1𝐴, 𝐵⦆ ∈ V)
31, 2eqeltrrid 2844 . . 3 (⦅𝐴, 𝐵⦆ ∈ V → 𝐴 ∈ V)
4 bj-pr22val 37002 . . . 4 pr2𝐴, 𝐵⦆ = 𝐵
5 bj-pr2ex 37003 . . . 4 (⦅𝐴, 𝐵⦆ ∈ V → pr2𝐴, 𝐵⦆ ∈ V)
64, 5eqeltrrid 2844 . . 3 (⦅𝐴, 𝐵⦆ ∈ V → 𝐵 ∈ V)
73, 6jca 511 . 2 (⦅𝐴, 𝐵⦆ ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V))
8 df-bj-2upl 36994 . . 3 𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
9 bj-1uplex 36991 . . . . 5 (⦅𝐴⦆ ∈ V ↔ 𝐴 ∈ V)
109biimpri 228 . . . 4 (𝐴 ∈ V → ⦅𝐴⦆ ∈ V)
11 snex 5442 . . . . 5 {1o} ∈ V
12 bj-xtagex 36972 . . . . 5 ({1o} ∈ V → (𝐵 ∈ V → ({1o} × tag 𝐵) ∈ V))
1311, 12ax-mp 5 . . . 4 (𝐵 ∈ V → ({1o} × tag 𝐵) ∈ V)
14 unexg 7762 . . . 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 2843 . 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 2106  Vcvv 3478  cun 3961  {csn 4631   × cxp 5687  1oc1o 8498  tag bj-ctag 36957  bj-c1upl 36980  pr1 bj-cpr1 36983  bj-c2uple 36993  pr2 bj-cpr2 36997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-suc 6392  df-1o 8505  df-bj-sngl 36949  df-bj-tag 36958  df-bj-proj 36974  df-bj-1upl 36981  df-bj-pr1 36984  df-bj-2upl 36994  df-bj-pr2 36998
This theorem is referenced by: (None)
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