Mathbox for BJ |
< Previous
Next >
Nearby theorems |
||
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 3416 | . . 3 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V) | |
2 | bj-tagex 34863 | . . 3 ⊢ (𝐵 ∈ V ↔ tag 𝐵 ∈ V) | |
3 | 1, 2 | sylib 221 | . 2 ⊢ (𝐵 ∈ 𝑊 → tag 𝐵 ∈ V) |
4 | bj-xpexg2 34836 | . 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 2112 Vcvv 3398 × cxp 5534 tag bj-ctag 34850 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-opab 5102 df-xp 5542 df-rel 5543 df-bj-sngl 34842 df-bj-tag 34851 |
This theorem is referenced by: bj-1uplex 34884 bj-2uplex 34898 |
Copyright terms: Public domain | W3C validator |