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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-2uplex | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
bj-2uplex | ⊢ (⦅𝐴, 𝐵⦆ ∈ V ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-pr21val 36996 | . . . 4 ⊢ pr1 ⦅𝐴, 𝐵⦆ = 𝐴 | |
2 | bj-pr1ex 36989 | . . . 4 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → pr1 ⦅𝐴, 𝐵⦆ ∈ V) | |
3 | 1, 2 | eqeltrrid 2844 | . . 3 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → 𝐴 ∈ V) |
4 | bj-pr22val 37002 | . . . 4 ⊢ pr2 ⦅𝐴, 𝐵⦆ = 𝐵 | |
5 | bj-pr2ex 37003 | . . . 4 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → pr2 ⦅𝐴, 𝐵⦆ ∈ V) | |
6 | 4, 5 | eqeltrrid 2844 | . . 3 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → 𝐵 ∈ V) |
7 | 3, 6 | jca 511 | . 2 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
8 | df-bj-2upl 36994 | . . 3 ⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) | |
9 | bj-1uplex 36991 | . . . . 5 ⊢ (⦅𝐴⦆ ∈ V ↔ 𝐴 ∈ V) | |
10 | 9 | biimpri 228 | . . . 4 ⊢ (𝐴 ∈ V → ⦅𝐴⦆ ∈ V) |
11 | snex 5442 | . . . . 5 ⊢ {1o} ∈ V | |
12 | bj-xtagex 36972 | . . . . 5 ⊢ ({1o} ∈ V → (𝐵 ∈ V → ({1o} × tag 𝐵) ∈ V)) | |
13 | 11, 12 | ax-mp 5 | . . . 4 ⊢ (𝐵 ∈ V → ({1o} × tag 𝐵) ∈ V) |
14 | unexg 7762 | . . . 4 ⊢ ((⦅𝐴⦆ ∈ V ∧ ({1o} × tag 𝐵) ∈ V) → (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) ∈ V) | |
15 | 10, 13, 14 | syl2an 596 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) ∈ V) |
16 | 8, 15 | eqeltrid 2843 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⦅𝐴, 𝐵⦆ ∈ V) |
17 | 7, 16 | impbii 209 | 1 ⊢ (⦅𝐴, 𝐵⦆ ∈ V ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2106 Vcvv 3478 ∪ cun 3961 {csn 4631 × cxp 5687 1oc1o 8498 tag bj-ctag 36957 ⦅bj-c1upl 36980 pr1 bj-cpr1 36983 ⦅bj-c2uple 36993 pr2 bj-cpr2 36997 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-suc 6392 df-1o 8505 df-bj-sngl 36949 df-bj-tag 36958 df-bj-proj 36974 df-bj-1upl 36981 df-bj-pr1 36984 df-bj-2upl 36994 df-bj-pr2 36998 |
This theorem is referenced by: (None) |
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