Theorem bj-2uplex 37774

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 37765	. . . 4 ⊢ pr1 ⦅𝐴, 𝐵⦆ = 𝐴
2		bj-pr1ex 37758	. . . 4 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → pr1 ⦅𝐴, 𝐵⦆ ∈ V)
3	1, 2	eqeltrrid 2867	. . 3 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → 𝐴 ∈ V)
4		bj-pr22val 37771	. . . 4 ⊢ pr2 ⦅𝐴, 𝐵⦆ = 𝐵
5		bj-pr2ex 37772	. . . 4 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → pr2 ⦅𝐴, 𝐵⦆ ∈ V)
6	4, 5	eqeltrrid 2867	. . 3 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → 𝐵 ∈ V)
7	3, 6	jca 521	. 2 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V))
8		df-bj-2upl 37763	. . 3 ⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
9		bj-1uplex 37760	. . . . 5 ⊢ (⦅𝐴⦆ ∈ V ↔ 𝐴 ∈ V)
10	9	biimpri 231	. . . 4 ⊢ (𝐴 ∈ V → ⦅𝐴⦆ ∈ V)
11		snex 5408	. . . . 5 ⊢ {1o} ∈ V
12		bj-xtagex 37741	. . . . 5 ⊢ ({1o} ∈ V → (𝐵 ∈ V → ({1o} × tag 𝐵) ∈ V))
13	11, 12	ax-mp 5	. . . 4 ⊢ (𝐵 ∈ V → ({1o} × tag 𝐵) ∈ V)
14		unexg 7749	. . . 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 2866	. 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 2145  Vcvv 3453   ∪ cun 3900  {csn 4587   × cxp 5657  1_oc1o 8452  tag bj-ctag 37726  ⦅bj-c1upl 37749  pr₁ bj-cpr1 37752  ⦅bj-c2uple 37762  pr₂ bj-cpr2 37766

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-xp 5665  df-rel 5666  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-suc 6367  df-1o 8459  df-bj-sngl 37718  df-bj-tag 37727  df-bj-proj 37743  df-bj-1upl 37750  df-bj-pr1 37753  df-bj-2upl 37763  df-bj-pr2 37767

This theorem is used by: (None)