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Theorem djussxp 5406
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 4695 . 2 ( 𝑥𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V) ↔ ∀𝑥𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V))
2 snssi 4474 . . 3 (𝑥𝐴 → {𝑥} ⊆ 𝐴)
3 ssv 3774 . . 3 𝐵 ⊆ V
4 xpss12 5264 . . 3 (({𝑥} ⊆ 𝐴𝐵 ⊆ V) → ({𝑥} × 𝐵) ⊆ (𝐴 × V))
52, 3, 4sylancl 574 . 2 (𝑥𝐴 → ({𝑥} × 𝐵) ⊆ (𝐴 × V))
61, 5mprgbir 3076 1 𝑥𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V)
Colors of variables: wff setvar class
Syntax hints:  wcel 2145  Vcvv 3351  wss 3723  {csn 4316   ciun 4654   × cxp 5247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-v 3353  df-in 3730  df-ss 3737  df-sn 4317  df-iun 4656  df-opab 4847  df-xp 5255
This theorem is referenced by:  djudisj  5702  iundom2g  9564
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