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Theorem bj-xtagex 36991
Description: The product of a set and the tagging of a set is a set. (Contributed by BJ, 2-Apr-2019.)
Assertion
Ref Expression
bj-xtagex (𝐴𝑉 → (𝐵𝑊 → (𝐴 × tag 𝐵) ∈ V))

Proof of Theorem bj-xtagex
StepHypRef Expression
1 elex 3500 . . 3 (𝐵𝑊𝐵 ∈ V)
2 bj-tagex 36989 . . 3 (𝐵 ∈ V ↔ tag 𝐵 ∈ V)
31, 2sylib 218 . 2 (𝐵𝑊 → tag 𝐵 ∈ V)
4 bj-xpexg2 36962 . 2 (𝐴𝑉 → (tag 𝐵 ∈ V → (𝐴 × tag 𝐵) ∈ V))
53, 4syl5 34 1 (𝐴𝑉 → (𝐵𝑊 → (𝐴 × tag 𝐵) ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  Vcvv 3479   × cxp 5682  tag bj-ctag 36976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-opab 5205  df-xp 5690  df-rel 5691  df-bj-sngl 36968  df-bj-tag 36977
This theorem is referenced by:  bj-1uplex  37010  bj-2uplex  37024
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