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Theorem djussxp 5710
 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 4961 . 2 ( 𝑥𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V) ↔ ∀𝑥𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V))
2 snssi 4734 . . 3 (𝑥𝐴 → {𝑥} ⊆ 𝐴)
3 ssv 3990 . . 3 𝐵 ⊆ V
4 xpss12 5564 . . 3 (({𝑥} ⊆ 𝐴𝐵 ⊆ V) → ({𝑥} × 𝐵) ⊆ (𝐴 × V))
52, 3, 4sylancl 588 . 2 (𝑥𝐴 → ({𝑥} × 𝐵) ⊆ (𝐴 × V))
61, 5mprgbir 3153 1 𝑥𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V)
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 2110  Vcvv 3494   ⊆ wss 3935  {csn 4560  ∪ ciun 4911   × cxp 5547 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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-v 3496  df-in 3942  df-ss 3951  df-sn 4561  df-iun 4913  df-opab 5121  df-xp 5555 This theorem is referenced by:  djudisj  6018  iundom2g  9956
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