Theorem elomssom 4729

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 4730. (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 3263	. 2 |- ( y = (/) -> ( y C_ om <-> (/) C_ om ) )
2		sseq1 3263	. 2 |- ( y = x -> ( y C_ om <-> x C_ om ) )
3		sseq1 3263	. 2 |- ( y = suc x -> ( y C_ om <-> suc x C_ om ) )
4		sseq1 3263	. 2 |- ( y = A -> ( y C_ om <-> A C_ om ) )
5		0ss 3549	. 2 |- (/) C_ om
6		unss 3395	. . . . 5 |- ( ( x C_ om /\ { x } C_ om ) <-> ( x u. { x } ) C_ om )
7		vex 2818	. . . . . . 7 |- x e. _V
8	7	snss 3831	. . . . . 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 4494	. . . . . 6 |- suc x = ( x u. { x } )
11	10	sseq1i 3266	. . . . 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 4724	1 |- ( A e. om -> A C_ om )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2205   u. cun 3211   C_ wss 3213   (/)c0 3510   {csn 3691   suc csuc 4488   omcom 4714

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-iinf 4712

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-uni 3917  df-int 3952  df-suc 4494  df-iom 4715

This theorem is referenced by:  elnn  4730  2ssom  6759  nninfwlpoimlemginf  7469  ennnfonelemg  13171