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Theorem djussxp 5844
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 5047 . 2 ( 𝑥𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V) ↔ ∀𝑥𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V))
2 snssi 4810 . . 3 (𝑥𝐴 → {𝑥} ⊆ 𝐴)
3 ssv 4005 . . 3 𝐵 ⊆ V
4 xpss12 5690 . . 3 (({𝑥} ⊆ 𝐴𝐵 ⊆ V) → ({𝑥} × 𝐵) ⊆ (𝐴 × V))
52, 3, 4sylancl 584 . 2 (𝑥𝐴 → ({𝑥} × 𝐵) ⊆ (𝐴 × V))
61, 5mprgbir 3066 1 𝑥𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V)
Colors of variables: wff setvar class
Syntax hints:  wcel 2104  Vcvv 3472  wss 3947  {csn 4627   ciun 4996   × cxp 5673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ral 3060  df-rex 3069  df-v 3474  df-in 3954  df-ss 3964  df-sn 4628  df-iun 4998  df-opab 5210  df-xp 5681
This theorem is referenced by:  djudisj  6165  iundom2g  10537
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