Theorem elomssom 4747

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 4748. (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 3271	. 2 |- ( y = (/) -> ( y C_ om <-> (/) C_ om ) )
2		sseq1 3271	. 2 |- ( y = x -> ( y C_ om <-> x C_ om ) )
3		sseq1 3271	. 2 |- ( y = suc x -> ( y C_ om <-> suc x C_ om ) )
4		sseq1 3271	. 2 |- ( y = A -> ( y C_ om <-> A C_ om ) )
5		0ss 3561	. 2 |- (/) C_ om
6		unss 3403	. . . . 5 |- ( ( x C_ om /\ { x } C_ om ) <-> ( x u. { x } ) C_ om )
7		vex 2824	. . . . . . 7 |- x e. _V
8	7	snss 3845	. . . . . 6 |- ( x e. om <-> { x } C_ om )
9	8	anbi2i 461	. . . . 5 |- ( ( x C_ om /\ x e. om ) <-> ( x C_ om /\ { x } C_ om ) )
10		df-suc 4511	. . . . . 6 |- suc x = ( x u. { x } )
11	10	sseq1i 3274	. . . . 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 4742	1 |- ( A e. om -> A C_ om )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2209   u. cun 3218   C_ wss 3220   (/)c0 3520   {csn 3705   suc csuc 4505   omcom 4732

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-uni 3931  df-int 3966  df-suc 4511  df-iom 4733

This theorem is referenced by:  elnn  4748  2ssom  6787  nninfwlpoimlemginf  7506  ennnfonelemg  13272