Theorem elomssom 4641

Description: A natural number ordinal is, as a set, included in the set of natural number ordinals. (Contributed by NM, 21-Jun-1998.) Extract this result from the previous proof of elnn 4642. (Revised by BJ, 7-Aug-2024.)

Assertion

Ref	Expression
elomssom	|- ( A e. om -> A C_ om )

Proof of Theorem elomssom

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseq1 3206	. 2 |- ( y = (/) -> ( y C_ om <-> (/) C_ om ) )
2		sseq1 3206	. 2 |- ( y = x -> ( y C_ om <-> x C_ om ) )
3		sseq1 3206	. 2 |- ( y = suc x -> ( y C_ om <-> suc x C_ om ) )
4		sseq1 3206	. 2 |- ( y = A -> ( y C_ om <-> A C_ om ) )
5		0ss 3489	. 2 |- (/) C_ om
6		unss 3337	. . . . 5 |- ( ( x C_ om /\ { x } C_ om ) <-> ( x u. { x } ) C_ om )
7		vex 2766	. . . . . . 7 |- x e. _V
8	7	snss 3757	. . . . . 6 |- ( x e. om <-> { x } C_ om )
9	8	anbi2i 457	. . . . 5 |- ( ( x C_ om /\ x e. om ) <-> ( x C_ om /\ { x } C_ om ) )
10		df-suc 4406	. . . . . 6 |- suc x = ( x u. { x } )
11	10	sseq1i 3209	. . . . 5 |- ( suc x C_ om <-> ( x u. { x } ) C_ om )
12	6, 9, 11	3bitr4i 212	. . . 4 |- ( ( x C_ om /\ x e. om ) <-> suc x C_ om )
13	12	biimpi 120	. . 3 |- ( ( x C_ om /\ x e. om ) -> suc x C_ om )
14	13	expcom 116	. 2 |- ( x e. om -> ( x C_ om -> suc x C_ om ) )
15	1, 2, 3, 4, 5, 14	finds 4636	1 |- ( A e. om -> A C_ om )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2167   u. cun 3155   C_ wss 3157   (/)c0 3450   {csn 3622   suc csuc 4400   omcom 4626

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-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-uni 3840  df-int 3875  df-suc 4406  df-iom 4627

This theorem is referenced by:  elnn  4642  2ssom  6582  nninfwlpoimlemginf  7242  ennnfonelemg  12620