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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. (Contributed by BJ, 6-Oct-2018.) |
Ref | Expression |
---|---|
bj-2uplex | ⊢ (⦅𝐴, 𝐵⦆ ∈ V ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-pr21val 33493 | . . . 4 ⊢ pr1 ⦅𝐴, 𝐵⦆ = 𝐴 | |
2 | bj-pr1ex 33486 | . . . 4 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → pr1 ⦅𝐴, 𝐵⦆ ∈ V) | |
3 | 1, 2 | syl5eqelr 2883 | . . 3 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → 𝐴 ∈ V) |
4 | bj-pr22val 33499 | . . . 4 ⊢ pr2 ⦅𝐴, 𝐵⦆ = 𝐵 | |
5 | bj-pr2ex 33500 | . . . 4 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → pr2 ⦅𝐴, 𝐵⦆ ∈ V) | |
6 | 4, 5 | syl5eqelr 2883 | . . 3 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → 𝐵 ∈ V) |
7 | 3, 6 | jca 508 | . 2 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
8 | df-bj-2upl 33491 | . . 3 ⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐵)) | |
9 | bj-1uplex 33488 | . . . . 5 ⊢ (⦅𝐴⦆ ∈ V ↔ 𝐴 ∈ V) | |
10 | 9 | biimpri 220 | . . . 4 ⊢ (𝐴 ∈ V → ⦅𝐴⦆ ∈ V) |
11 | snex 5099 | . . . . 5 ⊢ {1𝑜} ∈ V | |
12 | bj-xtagex 33469 | . . . . 5 ⊢ ({1𝑜} ∈ V → (𝐵 ∈ V → ({1𝑜} × tag 𝐵) ∈ V)) | |
13 | 11, 12 | ax-mp 5 | . . . 4 ⊢ (𝐵 ∈ V → ({1𝑜} × tag 𝐵) ∈ V) |
14 | unexg 7193 | . . . 4 ⊢ ((⦅𝐴⦆ ∈ V ∧ ({1𝑜} × tag 𝐵) ∈ V) → (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐵)) ∈ V) | |
15 | 10, 13, 14 | syl2an 590 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐵)) ∈ V) |
16 | 8, 15 | syl5eqel 2882 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⦅𝐴, 𝐵⦆ ∈ V) |
17 | 7, 16 | impbii 201 | 1 ⊢ (⦅𝐴, 𝐵⦆ ∈ V ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 ∈ wcel 2157 Vcvv 3385 ∪ cun 3767 {csn 4368 × cxp 5310 1𝑜c1o 7792 tag bj-ctag 33454 ⦅bj-c1upl 33477 pr1 bj-cpr1 33480 ⦅bj-c2uple 33490 pr2 bj-cpr2 33494 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-br 4844 df-opab 4906 df-tr 4946 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-ord 5944 df-on 5945 df-suc 5947 df-1o 7799 df-bj-sngl 33446 df-bj-tag 33455 df-bj-proj 33471 df-bj-1upl 33478 df-bj-pr1 33481 df-bj-2upl 33491 df-bj-pr2 33495 |
This theorem is referenced by: (None) |
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