Theorem bj-2uplex 36206

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

Step	Hyp	Ref	Expression
1		bj-pr21val 36197	. . . 4 ⊢ pr1 ⦅𝐴, 𝐵⦆ = 𝐴
2		bj-pr1ex 36190	. . . 4 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → pr1 ⦅𝐴, 𝐵⦆ ∈ V)
3	1, 2	eqeltrrid 2836	. . 3 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → 𝐴 ∈ V)
4		bj-pr22val 36203	. . . 4 ⊢ pr2 ⦅𝐴, 𝐵⦆ = 𝐵
5		bj-pr2ex 36204	. . . 4 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → pr2 ⦅𝐴, 𝐵⦆ ∈ V)
6	4, 5	eqeltrrid 2836	. . 3 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → 𝐵 ∈ V)
7	3, 6	jca 510	. 2 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V))
8		df-bj-2upl 36195	. . 3 ⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
9		bj-1uplex 36192	. . . . 5 ⊢ (⦅𝐴⦆ ∈ V ↔ 𝐴 ∈ V)
10	9	biimpri 227	. . . 4 ⊢ (𝐴 ∈ V → ⦅𝐴⦆ ∈ V)
11		snex 5430	. . . . 5 ⊢ {1o} ∈ V
12		bj-xtagex 36173	. . . . 5 ⊢ ({1o} ∈ V → (𝐵 ∈ V → ({1o} × tag 𝐵) ∈ V))
13	11, 12	ax-mp 5	. . . 4 ⊢ (𝐵 ∈ V → ({1o} × tag 𝐵) ∈ V)
14		unexg 7738	. . . 4 ⊢ ((⦅𝐴⦆ ∈ V ∧ ({1o} × tag 𝐵) ∈ V) → (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) ∈ V)
15	10, 13, 14	syl2an 594	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) ∈ V)
16	8, 15	eqeltrid 2835	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⦅𝐴, 𝐵⦆ ∈ V)
17	7, 16	impbii 208	1 ⊢ (⦅𝐴, 𝐵⦆ ∈ V ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∈ wcel 2104  Vcvv 3472   ∪ cun 3945  {csn 4627   × cxp 5673  1_oc1o 8461  tag bj-ctag 36158  ⦅bj-c1upl 36181  pr₁ bj-cpr1 36184  ⦅bj-c2uple 36194  pr₂ bj-cpr2 36198

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682  df-cnv 5683  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-suc 6369  df-1o 8468  df-bj-sngl 36150  df-bj-tag 36159  df-bj-proj 36175  df-bj-1upl 36182  df-bj-pr1 36185  df-bj-2upl 36195  df-bj-pr2 36199

This theorem is referenced by: (None)