Theorem bj-2uplex 38027

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 38018	. . . 4 ⊢ pr1 ⦅𝐴, 𝐵⦆ = 𝐴
2		bj-pr1ex 38011	. . . 4 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → pr1 ⦅𝐴, 𝐵⦆ ∈ V)
3	1, 2	eqeltrrid 2865	. . 3 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → 𝐴 ∈ V)
4		bj-pr22val 38024	. . . 4 ⊢ pr2 ⦅𝐴, 𝐵⦆ = 𝐵
5		bj-pr2ex 38025	. . . 4 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → pr2 ⦅𝐴, 𝐵⦆ ∈ V)
6	4, 5	eqeltrrid 2865	. . 3 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → 𝐵 ∈ V)
7	3, 6	jca 521	. 2 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V))
8		df-bj-2upl 38016	. . 3 ⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
9		bj-1uplex 38013	. . . . 5 ⊢ (⦅𝐴⦆ ∈ V ↔ 𝐴 ∈ V)
10	9	biimpri 231	. . . 4 ⊢ (𝐴 ∈ V → ⦅𝐴⦆ ∈ V)
11		snex 5396	. . . . 5 ⊢ {1o} ∈ V
12		bj-xtagex 37994	. . . . 5 ⊢ ({1o} ∈ V → (𝐵 ∈ V → ({1o} × tag 𝐵) ∈ V))
13	11, 12	ax-mp 5	. . . 4 ⊢ (𝐵 ∈ V → ({1o} × tag 𝐵) ∈ V)
14		unexg 7760	. . . 4 ⊢ ((⦅𝐴⦆ ∈ V ∧ ({1o} × tag 𝐵) ∈ V) → (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) ∈ V)
15	10, 13, 14	syl2an 608	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) ∈ V)
16	8, 15	eqeltrid 2864	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⦅𝐴, 𝐵⦆ ∈ V)
17	7, 16	impbii 212	1 ⊢ (⦅𝐴, 𝐵⦆ ∈ V ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2144  Vcvv 3450   ∪ cun 4149  {csn 4584   × cxp 5648  1_oc1o 8472  tag bj-ctag 37979  ⦅bj-c1upl 38002  pr₁ bj-cpr1 38005  ⦅bj-c2uple 38015  pr₂ bj-cpr2 38019

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-xp 5656  df-rel 5657  df-cnv 5658  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-suc 6368  df-1o 8479  df-bj-sngl 37971  df-bj-tag 37980  df-bj-proj 37996  df-bj-1upl 38003  df-bj-pr1 38006  df-bj-2upl 38016  df-bj-pr2 38020

This theorem is used by: (None)