Theorem elomssom 4654

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 4655. (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 3216	. 2 |- ( y = (/) -> ( y C_ om <-> (/) C_ om ) )
2		sseq1 3216	. 2 |- ( y = x -> ( y C_ om <-> x C_ om ) )
3		sseq1 3216	. 2 |- ( y = suc x -> ( y C_ om <-> suc x C_ om ) )
4		sseq1 3216	. 2 |- ( y = A -> ( y C_ om <-> A C_ om ) )
5		0ss 3499	. 2 |- (/) C_ om
6		unss 3347	. . . . 5 |- ( ( x C_ om /\ { x } C_ om ) <-> ( x u. { x } ) C_ om )
7		vex 2775	. . . . . . 7 |- x e. _V
8	7	snss 3768	. . . . . 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 4419	. . . . . 6 |- suc x = ( x u. { x } )
11	10	sseq1i 3219	. . . . 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 4649	1 |- ( A e. om -> A C_ om )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2176   u. cun 3164   C_ wss 3166   (/)c0 3460   {csn 3633   suc csuc 4413   omcom 4639

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-iinf 4637

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-uni 3851  df-int 3886  df-suc 4419  df-iom 4640

This theorem is referenced by:  elnn  4655  2ssom  6612  nninfwlpoimlemginf  7280  ennnfonelemg  12807