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Theorem bj-xtagex 36972
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 3499 . . 3 (𝐵𝑊𝐵 ∈ V)
2 bj-tagex 36970 . . 3 (𝐵 ∈ V ↔ tag 𝐵 ∈ V)
31, 2sylib 218 . 2 (𝐵𝑊 → tag 𝐵 ∈ V)
4 bj-xpexg2 36943 . 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 2106  Vcvv 3478   × cxp 5687  tag bj-ctag 36957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-opab 5211  df-xp 5695  df-rel 5696  df-bj-sngl 36949  df-bj-tag 36958
This theorem is referenced by:  bj-1uplex  36991  bj-2uplex  37005
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