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Theorem bj-xtagex 33102
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 3243 . . 3 (𝐵𝑊𝐵 ∈ V)
2 bj-tagex 33100 . . 3 (𝐵 ∈ V ↔ tag 𝐵 ∈ V)
31, 2sylib 208 . 2 (𝐵𝑊 → tag 𝐵 ∈ V)
4 bj-xpexg2 33072 . 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 2030  Vcvv 3231   × cxp 5141  tag bj-ctag 33087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-opab 4746  df-xp 5149  df-rel 5150  df-bj-sngl 33079  df-bj-tag 33088
This theorem is referenced by:  bj-1uplex  33121  bj-2uplex  33135
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