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Theorem ssonunii 7134
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 7133 . 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 2145  Vcvv 3351  wss 3723   cuni 4574  Oncon0 5866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-tr 4887  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-ord 5869  df-on 5870
This theorem is referenced by:  bm2.5ii  7153  tz9.12lem2  8815  ttukeylem6  9538  onsetreclem2  42980
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