Theorem bj-2uplex 36506

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 36497	. . . 4 ⊢ pr1 ⦅𝐴, 𝐵⦆ = 𝐴
2		bj-pr1ex 36490	. . . 4 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → pr1 ⦅𝐴, 𝐵⦆ ∈ V)
3	1, 2	eqeltrrid 2833	. . 3 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → 𝐴 ∈ V)
4		bj-pr22val 36503	. . . 4 ⊢ pr2 ⦅𝐴, 𝐵⦆ = 𝐵
5		bj-pr2ex 36504	. . . 4 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → pr2 ⦅𝐴, 𝐵⦆ ∈ V)
6	4, 5	eqeltrrid 2833	. . 3 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → 𝐵 ∈ V)
7	3, 6	jca 510	. 2 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V))
8		df-bj-2upl 36495	. . 3 ⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
9		bj-1uplex 36492	. . . . 5 ⊢ (⦅𝐴⦆ ∈ V ↔ 𝐴 ∈ V)
10	9	biimpri 227	. . . 4 ⊢ (𝐴 ∈ V → ⦅𝐴⦆ ∈ V)
11		snex 5435	. . . . 5 ⊢ {1o} ∈ V
12		bj-xtagex 36473	. . . . 5 ⊢ ({1o} ∈ V → (𝐵 ∈ V → ({1o} × tag 𝐵) ∈ V))
13	11, 12	ax-mp 5	. . . 4 ⊢ (𝐵 ∈ V → ({1o} × tag 𝐵) ∈ V)
14		unexg 7755	. . . 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 2832	. 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 2098  Vcvv 3471   ∪ cun 3945  {csn 4630   × cxp 5678  1_oc1o 8484  tag bj-ctag 36458  ⦅bj-c1upl 36481  pr₁ bj-cpr1 36484  ⦅bj-c2uple 36494  pr₂ bj-cpr2 36498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-xp 5686  df-rel 5687  df-cnv 5688  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-suc 6378  df-1o 8491  df-bj-sngl 36450  df-bj-tag 36459  df-bj-proj 36475  df-bj-1upl 36482  df-bj-pr1 36485  df-bj-2upl 36495  df-bj-pr2 36499

This theorem is referenced by: (None)