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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 34328 | . . . 4 ⊢ pr1 ⦅𝐴, 𝐵⦆ = 𝐴 | |
2 | bj-pr1ex 34321 | . . . 4 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → pr1 ⦅𝐴, 𝐵⦆ ∈ V) | |
3 | 1, 2 | eqeltrrid 2918 | . . 3 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → 𝐴 ∈ V) |
4 | bj-pr22val 34334 | . . . 4 ⊢ pr2 ⦅𝐴, 𝐵⦆ = 𝐵 | |
5 | bj-pr2ex 34335 | . . . 4 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → pr2 ⦅𝐴, 𝐵⦆ ∈ V) | |
6 | 4, 5 | eqeltrrid 2918 | . . 3 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → 𝐵 ∈ V) |
7 | 3, 6 | jca 514 | . 2 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
8 | df-bj-2upl 34326 | . . 3 ⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) | |
9 | bj-1uplex 34323 | . . . . 5 ⊢ (⦅𝐴⦆ ∈ V ↔ 𝐴 ∈ V) | |
10 | 9 | biimpri 230 | . . . 4 ⊢ (𝐴 ∈ V → ⦅𝐴⦆ ∈ V) |
11 | snex 5332 | . . . . 5 ⊢ {1o} ∈ V | |
12 | bj-xtagex 34304 | . . . . 5 ⊢ ({1o} ∈ V → (𝐵 ∈ V → ({1o} × tag 𝐵) ∈ V)) | |
13 | 11, 12 | ax-mp 5 | . . . 4 ⊢ (𝐵 ∈ V → ({1o} × tag 𝐵) ∈ V) |
14 | unexg 7472 | . . . 4 ⊢ ((⦅𝐴⦆ ∈ V ∧ ({1o} × tag 𝐵) ∈ V) → (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) ∈ V) | |
15 | 10, 13, 14 | syl2an 597 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) ∈ V) |
16 | 8, 15 | eqeltrid 2917 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⦅𝐴, 𝐵⦆ ∈ V) |
17 | 7, 16 | impbii 211 | 1 ⊢ (⦅𝐴, 𝐵⦆ ∈ V ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2114 Vcvv 3494 ∪ cun 3934 {csn 4567 × cxp 5553 1oc1o 8095 tag bj-ctag 34289 ⦅bj-c1upl 34312 pr1 bj-cpr1 34315 ⦅bj-c2uple 34325 pr2 bj-cpr2 34329 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-tr 5173 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-ord 6194 df-on 6195 df-suc 6197 df-1o 8102 df-bj-sngl 34281 df-bj-tag 34290 df-bj-proj 34306 df-bj-1upl 34313 df-bj-pr1 34316 df-bj-2upl 34326 df-bj-pr2 34330 |
This theorem is referenced by: (None) |
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