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Theorem bj-xtagex 34426
 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 3462 . . 3 (𝐵𝑊𝐵 ∈ V)
2 bj-tagex 34424 . . 3 (𝐵 ∈ V ↔ tag 𝐵 ∈ V)
31, 2sylib 221 . 2 (𝐵𝑊 → tag 𝐵 ∈ V)
4 bj-xpexg2 34397 . 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 2112  Vcvv 3444   × cxp 5521  tag bj-ctag 34411 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-opab 5096  df-xp 5529  df-rel 5530  df-bj-sngl 34403  df-bj-tag 34412 This theorem is referenced by:  bj-1uplex  34445  bj-2uplex  34459
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