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Theorem djussxp 5471
Description: Disjoint union is a subset of a Cartesian product. (Contributed by Stefan O'Rear, 21-Nov-2014.)
Assertion
Ref Expression
djussxp 𝑥𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem djussxp
StepHypRef Expression
1 iunss 4751 . 2 ( 𝑥𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V) ↔ ∀𝑥𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V))
2 snssi 4527 . . 3 (𝑥𝐴 → {𝑥} ⊆ 𝐴)
3 ssv 3821 . . 3 𝐵 ⊆ V
4 xpss12 5327 . . 3 (({𝑥} ⊆ 𝐴𝐵 ⊆ V) → ({𝑥} × 𝐵) ⊆ (𝐴 × V))
52, 3, 4sylancl 581 . 2 (𝑥𝐴 → ({𝑥} × 𝐵) ⊆ (𝐴 × V))
61, 5mprgbir 3108 1 𝑥𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V)
Colors of variables: wff setvar class
Syntax hints:  wcel 2157  Vcvv 3385  wss 3769  {csn 4368   ciun 4710   × cxp 5310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-v 3387  df-in 3776  df-ss 3783  df-sn 4369  df-iun 4712  df-opab 4906  df-xp 5318
This theorem is referenced by:  djudisj  5778  iundom2g  9650
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