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Theorem ssonunii 7789
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 7788 . 2 (𝐴 ∈ V → (𝐴 ⊆ On → 𝐴 ∈ On))
31, 2ax-mp 5 1 (𝐴 ⊆ On → 𝐴 ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  Vcvv 3473  wss 3949   cuni 4912  Oncon0 6374
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-tr 5270  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-ord 6377  df-on 6378
This theorem is referenced by:  uniordint  7810  tz9.12lem2  9819  ttukeylem6  10545  onsetreclem2  48215
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