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Theorem uniss2 3706
Description: A subclass condition on the members of two classes that implies a subclass relation on their unions. Proposition 8.6 of [TakeutiZaring] p. 59. (Contributed by NM, 22-Mar-2004.)
Assertion
Ref Expression
uniss2 (∀𝑥𝐴𝑦𝐵 𝑥𝑦 𝐴 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝐵
Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem uniss2
StepHypRef Expression
1 ssuni 3697 . . . . 5 ((𝑥𝑦𝑦𝐵) → 𝑥 𝐵)
21expcom 115 . . . 4 (𝑦𝐵 → (𝑥𝑦𝑥 𝐵))
32rexlimiv 2496 . . 3 (∃𝑦𝐵 𝑥𝑦𝑥 𝐵)
43ralimi 2449 . 2 (∀𝑥𝐴𝑦𝐵 𝑥𝑦 → ∀𝑥𝐴 𝑥 𝐵)
5 unissb 3705 . 2 ( 𝐴 𝐵 ↔ ∀𝑥𝐴 𝑥 𝐵)
64, 5sylibr 133 1 (∀𝑥𝐴𝑦𝐵 𝑥𝑦 𝐴 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1445  wral 2370  wrex 2371  wss 3013   cuni 3675
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-in 3019  df-ss 3026  df-uni 3676
This theorem is referenced by:  unidif  3707
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