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Theorem djussxp2 30413
 Description: Stronger version of djussxp 5684 (Contributed by Thierry Arnoux, 23-Jun-2024.)
Assertion
Ref Expression
djussxp2 𝑘𝐴 ({𝑘} × 𝐵) ⊆ (𝐴 × 𝑘𝐴 𝐵)
Distinct variable group:   𝐴,𝑘
Allowed substitution hint:   𝐵(𝑘)

Proof of Theorem djussxp2
StepHypRef Expression
1 nfcv 2958 . . . 4 𝑘𝐴
2 nfiu1 4918 . . . 4 𝑘 𝑘𝐴 𝐵
31, 2nfxp 5556 . . 3 𝑘(𝐴 × 𝑘𝐴 𝐵)
43iunssf 4934 . 2 ( 𝑘𝐴 ({𝑘} × 𝐵) ⊆ (𝐴 × 𝑘𝐴 𝐵) ↔ ∀𝑘𝐴 ({𝑘} × 𝐵) ⊆ (𝐴 × 𝑘𝐴 𝐵))
5 snssi 4704 . . 3 (𝑘𝐴 → {𝑘} ⊆ 𝐴)
6 ssiun2 4937 . . 3 (𝑘𝐴𝐵 𝑘𝐴 𝐵)
7 xpss12 5538 . . 3 (({𝑘} ⊆ 𝐴𝐵 𝑘𝐴 𝐵) → ({𝑘} × 𝐵) ⊆ (𝐴 × 𝑘𝐴 𝐵))
85, 6, 7syl2anc 587 . 2 (𝑘𝐴 → ({𝑘} × 𝐵) ⊆ (𝐴 × 𝑘𝐴 𝐵))
94, 8mprgbir 3124 1 𝑘𝐴 ({𝑘} × 𝐵) ⊆ (𝐴 × 𝑘𝐴 𝐵)
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 2112   ⊆ wss 3884  {csn 4528  ∪ ciun 4884   × cxp 5521 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 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-in 3891  df-ss 3901  df-sn 4529  df-iun 4886  df-opab 5096  df-xp 5529 This theorem is referenced by:  2ndresdju  30414  gsumpart  30743
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