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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-xtagex | Structured version Visualization version GIF version |
Description: The product of a set and the tagging of a set is a set. (Contributed by BJ, 2-Apr-2019.) |
Ref | Expression |
---|---|
bj-xtagex | ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ 𝑊 → (𝐴 × tag 𝐵) ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3489 | . . 3 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V) | |
2 | bj-tagex 36461 | . . 3 ⊢ (𝐵 ∈ V ↔ tag 𝐵 ∈ V) | |
3 | 1, 2 | sylib 217 | . 2 ⊢ (𝐵 ∈ 𝑊 → tag 𝐵 ∈ V) |
4 | bj-xpexg2 36434 | . 2 ⊢ (𝐴 ∈ 𝑉 → (tag 𝐵 ∈ V → (𝐴 × tag 𝐵) ∈ V)) | |
5 | 3, 4 | syl5 34 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ 𝑊 → (𝐴 × tag 𝐵) ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 Vcvv 3470 × cxp 5671 tag bj-ctag 36448 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-opab 5206 df-xp 5679 df-rel 5680 df-bj-sngl 36440 df-bj-tag 36449 |
This theorem is referenced by: bj-1uplex 36482 bj-2uplex 36496 |
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