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Theorem djussxp 4691
Description: Disjoint union is a subset of a cross 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 3861 . 2 ( 𝑥𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V) ↔ ∀𝑥𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V))
2 snssi 3671 . . 3 (𝑥𝐴 → {𝑥} ⊆ 𝐴)
3 ssv 3123 . . 3 𝐵 ⊆ V
4 xpss12 4653 . . 3 (({𝑥} ⊆ 𝐴𝐵 ⊆ V) → ({𝑥} × 𝐵) ⊆ (𝐴 × V))
52, 3, 4sylancl 410 . 2 (𝑥𝐴 → ({𝑥} × 𝐵) ⊆ (𝐴 × V))
61, 5mprgbir 2493 1 𝑥𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V)
Colors of variables: wff set class
Syntax hints:  wcel 1481  Vcvv 2689  wss 3075  {csn 3531   ciun 3820   × cxp 4544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-in 3081  df-ss 3088  df-sn 3537  df-iun 3822  df-opab 3997  df-xp 4552
This theorem is referenced by:  djudisj  4973
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