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Theorem unss 3310
Description: The union of two subclasses is a subclass. Theorem 27 of [Suppes] p. 27 and its converse. (Contributed by NM, 11-Jun-2004.)
Assertion
Ref Expression
unss ((𝐴𝐶𝐵𝐶) ↔ (𝐴𝐵) ⊆ 𝐶)

Proof of Theorem unss
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dfss2 3145 . 2 ((𝐴𝐵) ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ (𝐴𝐵) → 𝑥𝐶))
2 19.26 1481 . . 3 (∀𝑥((𝑥𝐴𝑥𝐶) ∧ (𝑥𝐵𝑥𝐶)) ↔ (∀𝑥(𝑥𝐴𝑥𝐶) ∧ ∀𝑥(𝑥𝐵𝑥𝐶)))
3 elun 3277 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
43imbi1i 238 . . . . 5 ((𝑥 ∈ (𝐴𝐵) → 𝑥𝐶) ↔ ((𝑥𝐴𝑥𝐵) → 𝑥𝐶))
5 jaob 710 . . . . 5 (((𝑥𝐴𝑥𝐵) → 𝑥𝐶) ↔ ((𝑥𝐴𝑥𝐶) ∧ (𝑥𝐵𝑥𝐶)))
64, 5bitri 184 . . . 4 ((𝑥 ∈ (𝐴𝐵) → 𝑥𝐶) ↔ ((𝑥𝐴𝑥𝐶) ∧ (𝑥𝐵𝑥𝐶)))
76albii 1470 . . 3 (∀𝑥(𝑥 ∈ (𝐴𝐵) → 𝑥𝐶) ↔ ∀𝑥((𝑥𝐴𝑥𝐶) ∧ (𝑥𝐵𝑥𝐶)))
8 dfss2 3145 . . . 4 (𝐴𝐶 ↔ ∀𝑥(𝑥𝐴𝑥𝐶))
9 dfss2 3145 . . . 4 (𝐵𝐶 ↔ ∀𝑥(𝑥𝐵𝑥𝐶))
108, 9anbi12i 460 . . 3 ((𝐴𝐶𝐵𝐶) ↔ (∀𝑥(𝑥𝐴𝑥𝐶) ∧ ∀𝑥(𝑥𝐵𝑥𝐶)))
112, 7, 103bitr4i 212 . 2 (∀𝑥(𝑥 ∈ (𝐴𝐵) → 𝑥𝐶) ↔ (𝐴𝐶𝐵𝐶))
121, 11bitr2i 185 1 ((𝐴𝐶𝐵𝐶) ↔ (𝐴𝐵) ⊆ 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wo 708  wal 1351  wcel 2148  cun 3128  wss 3130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-un 3134  df-in 3136  df-ss 3143
This theorem is referenced by:  unssi  3311  unssd  3312  unssad  3313  unssbd  3314  uneqin  3387  undifss  3504  prss  3749  prssg  3750  tpss  3759  exmid1stab  4209  pwundifss  4286  ordsucss  4504  elomssom  4605  eqrelrel  4728  xpsspw  4739  relun  4744  relcoi2  5160  dfer2  6536  fimaxre2  11235  uncld  13616  bdeqsuc  14636
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