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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 3515 | . . 3 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V) | |
2 | bj-tagex 34303 | . . 3 ⊢ (𝐵 ∈ V ↔ tag 𝐵 ∈ V) | |
3 | 1, 2 | sylib 220 | . 2 ⊢ (𝐵 ∈ 𝑊 → tag 𝐵 ∈ V) |
4 | bj-xpexg2 34276 | . 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 2113 Vcvv 3497 × cxp 5556 tag bj-ctag 34290 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-opab 5132 df-xp 5564 df-rel 5565 df-bj-sngl 34282 df-bj-tag 34291 |
This theorem is referenced by: bj-1uplex 34324 bj-2uplex 34338 |
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