Theorem elomssom 4701

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 4702. (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 3248	. 2 |- ( y = (/) -> ( y C_ om <-> (/) C_ om ) )
2		sseq1 3248	. 2 |- ( y = x -> ( y C_ om <-> x C_ om ) )
3		sseq1 3248	. 2 |- ( y = suc x -> ( y C_ om <-> suc x C_ om ) )
4		sseq1 3248	. 2 |- ( y = A -> ( y C_ om <-> A C_ om ) )
5		0ss 3531	. 2 |- (/) C_ om
6		unss 3379	. . . . 5 |- ( ( x C_ om /\ { x } C_ om ) <-> ( x u. { x } ) C_ om )
7		vex 2803	. . . . . . 7 |- x e. _V
8	7	snss 3806	. . . . . 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 4466	. . . . . 6 |- suc x = ( x u. { x } )
11	10	sseq1i 3251	. . . . 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 4696	1 |- ( A e. om -> A C_ om )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2200   u. cun 3196   C_ wss 3198   (/)c0 3492   {csn 3667   suc csuc 4460   omcom 4686

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-uni 3892  df-int 3927  df-suc 4466  df-iom 4687

This theorem is referenced by:  elnn  4702  2ssom  6687  nninfwlpoimlemginf  7366  ennnfonelemg  13014