Theorem ordsson 4503

Description: Any ordinal class is a subclass of the class of ordinal numbers. Corollary 7.15 of [TakeutiZaring] p. 38. (Contributed by NM, 18-May-1994.)

Assertion

Ref	Expression
ordsson	⊢ (Ord 𝐴 → 𝐴 ⊆ On)

Proof of Theorem ordsson

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordelon 4395	. . 3 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ On)
2	1	ex 115	. 2 ⊢ (Ord 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ∈ On))
3	2	ssrdv 3173	1 ⊢ (Ord 𝐴 → 𝐴 ⊆ On)

Syntax hints:   → wi 4   ∈ wcel 2158   ⊆ wss 3141  Ord word 4374  Oncon0 4375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-in 3147  df-ss 3154  df-uni 3822  df-tr 4114  df-iord 4378  df-on 4380

This theorem is referenced by:  onss  4504  orduni  4506  iordsmo  6312  tfrlemi14d  6348  tfr1onlemssrecs  6354  tfri1dALT  6366  tfrcllemssrecs  6367  ordiso2  7048