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Theorem onssi 7272
Description: An ordinal number is a subset of On. (Contributed by NM, 11-Aug-1994.)
Hypothesis
Ref Expression
onssi.1 𝐴 ∈ On
Assertion
Ref Expression
onssi 𝐴 ⊆ On

Proof of Theorem onssi
StepHypRef Expression
1 onssi.1 . 2 𝐴 ∈ On
2 onss 7225 . 2 (𝐴 ∈ On → 𝐴 ⊆ On)
31, 2ax-mp 5 1 𝐴 ⊆ On
Colors of variables: wff setvar class
Syntax hints:  wcel 2157  wss 3770  Oncon0 5942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pr 5098  ax-un 7184
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-pss 3786  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-tp 4374  df-op 4376  df-uni 4630  df-br 4845  df-opab 4907  df-tr 4947  df-eprel 5226  df-po 5234  df-so 5235  df-fr 5272  df-we 5274  df-ord 5945  df-on 5946
This theorem is referenced by:  rankbnd2  8983  dfac12r  9257  cfsmolem  9381  ttukeylem6  9625
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