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Theorem ssonuni 7103
Description: The union of a set of ordinal numbers is an ordinal number. Theorem 9 of [Suppes] p. 132. (Contributed by NM, 1-Nov-2003.)
Assertion
Ref Expression
ssonuni (𝐴𝑉 → (𝐴 ⊆ On → 𝐴 ∈ On))

Proof of Theorem ssonuni
StepHypRef Expression
1 ssorduni 7102 . 2 (𝐴 ⊆ On → Ord 𝐴)
2 uniexg 7072 . . 3 (𝐴𝑉 𝐴 ∈ V)
3 elong 5844 . . 3 ( 𝐴 ∈ V → ( 𝐴 ∈ On ↔ Ord 𝐴))
42, 3syl 17 . 2 (𝐴𝑉 → ( 𝐴 ∈ On ↔ Ord 𝐴))
51, 4syl5ibr 236 1 (𝐴𝑉 → (𝐴 ⊆ On → 𝐴 ∈ On))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wcel 2103  Vcvv 3304  wss 3680   cuni 4544  Ord word 5835  Oncon0 5836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-tr 4861  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-ord 5839  df-on 5840
This theorem is referenced by:  ssonunii  7104  onuni  7110  iunon  7556  onfununi  7558  oemapvali  8694  cardprclem  8918  carduni  8920  dfac12lem2  9079  ontgval  32657
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