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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 3499 | . . 3 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V) | |
2 | bj-tagex 36970 | . . 3 ⊢ (𝐵 ∈ V ↔ tag 𝐵 ∈ V) | |
3 | 1, 2 | sylib 218 | . 2 ⊢ (𝐵 ∈ 𝑊 → tag 𝐵 ∈ V) |
4 | bj-xpexg2 36943 | . 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 2106 Vcvv 3478 × cxp 5687 tag bj-ctag 36957 |
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-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-opab 5211 df-xp 5695 df-rel 5696 df-bj-sngl 36949 df-bj-tag 36958 |
This theorem is referenced by: bj-1uplex 36991 bj-2uplex 37005 |
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