Theorem onssi 7814

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 7764	. 2 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ⊆ On

Syntax hints:   ∈ wcel 2141   ⊆ wss 3904  Oncon0 6342

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-tr 5207  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-ord 6345  df-on 6346

This theorem is referenced by:  rankbnd2  9824  dfac12r  10100  cfsmolem  10224  ttukeylem6  10468