Theorem bj-2uplex 37376

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 37367	. . . 4 ⊢ pr1 ⦅𝐴, 𝐵⦆ = 𝐴
2		bj-pr1ex 37360	. . . 4 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → pr1 ⦅𝐴, 𝐵⦆ ∈ V)
3	1, 2	eqeltrrid 2845	. . 3 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → 𝐴 ∈ V)
4		bj-pr22val 37373	. . . 4 ⊢ pr2 ⦅𝐴, 𝐵⦆ = 𝐵
5		bj-pr2ex 37374	. . . 4 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → pr2 ⦅𝐴, 𝐵⦆ ∈ V)
6	4, 5	eqeltrrid 2845	. . 3 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → 𝐵 ∈ V)
7	3, 6	jca 516	. 2 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V))
8		df-bj-2upl 37365	. . 3 ⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
9		bj-1uplex 37362	. . . . 5 ⊢ (⦅𝐴⦆ ∈ V ↔ 𝐴 ∈ V)
10	9	biimpri 229	. . . 4 ⊢ (𝐴 ∈ V → ⦅𝐴⦆ ∈ V)
11		snex 5375	. . . . 5 ⊢ {1o} ∈ V
12		bj-xtagex 37343	. . . . 5 ⊢ ({1o} ∈ V → (𝐵 ∈ V → ({1o} × tag 𝐵) ∈ V))
13	11, 12	ax-mp 5	. . . 4 ⊢ (𝐵 ∈ V → ({1o} × tag 𝐵) ∈ V)
14		unexg 7693	. . . 4 ⊢ ((⦅𝐴⦆ ∈ V ∧ ({1o} × tag 𝐵) ∈ V) → (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) ∈ V)
15	10, 13, 14	syl2an 602	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) ∈ V)
16	8, 15	eqeltrid 2844	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⦅𝐴, 𝐵⦆ ∈ V)
17	7, 16	impbii 210	1 ⊢ (⦅𝐴, 𝐵⦆ ∈ V ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∈ wcel 2119  Vcvv 3432   ∪ cun 3888  {csn 4562   × cxp 5623  1_oc1o 8395  tag bj-ctag 37328  ⦅bj-c1upl 37351  pr₁ bj-cpr1 37354  ⦅bj-c2uple 37364  pr₂ bj-cpr2 37368

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-xp 5631  df-rel 5632  df-cnv 5633  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-suc 6323  df-1o 8402  df-bj-sngl 37320  df-bj-tag 37329  df-bj-proj 37345  df-bj-1upl 37352  df-bj-pr1 37355  df-bj-2upl 37365  df-bj-pr2 37369

This theorem is referenced by: (None)