Theorem onssi 7571

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

Step	Hyp	Ref	Expression
1		onssi.1	. 2 ⊢ 𝐴 ∈ On
2		onss 7524	. 2 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ⊆ On

Syntax hints:   ∈ wcel 2114   ⊆ wss 3843  Oncon0 6172

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-11 2162  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296  ax-un 7479

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-tr 5137  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-ord 6175  df-on 6176

This theorem is referenced by:  rankbnd2  9371  dfac12r  9646  cfsmolem  9770  ttukeylem6  10014