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Theorem bj-xtagex 36525
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 3482 . . 3 (𝐵𝑊𝐵 ∈ V)
2 bj-tagex 36523 . . 3 (𝐵 ∈ V ↔ tag 𝐵 ∈ V)
31, 2sylib 217 . 2 (𝐵𝑊 → tag 𝐵 ∈ V)
4 bj-xpexg2 36496 . 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 2098  Vcvv 3463   × cxp 5670  tag bj-ctag 36510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-opab 5206  df-xp 5678  df-rel 5679  df-bj-sngl 36502  df-bj-tag 36511
This theorem is referenced by:  bj-1uplex  36544  bj-2uplex  36558
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