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Theorem ssonunii 4484
Description: The union of a set of ordinal numbers is an ordinal number. Corollary 7N(d) of [Enderton] p. 193. (Contributed by NM, 20-Sep-2003.)
Hypothesis
Ref Expression
ssonuni.1 𝐴 ∈ V
Assertion
Ref Expression
ssonunii (𝐴 ⊆ On → 𝐴 ∈ On)

Proof of Theorem ssonunii
StepHypRef Expression
1 ssonuni.1 . 2 𝐴 ∈ V
2 ssonuni 4483 . 2 (𝐴 ∈ V → (𝐴 ⊆ On → 𝐴 ∈ On))
31, 2ax-mp 5 1 (𝐴 ⊆ On → 𝐴 ∈ On)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2148  Vcvv 2737  wss 3129   cuni 3807  Oncon0 4359
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-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-un 4429
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-in 3135  df-ss 3142  df-uni 3808  df-tr 4099  df-iord 4362  df-on 4364
This theorem is referenced by:  bm2.5ii  4491
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