Theorem bj-omtrans 10081

Description: The set om is transitive. A natural number is included in om. Constructive proof of elnn 4328.

The idea is to use bounded induction with the formula x C_ om. This formula, in a logic with terms, is bounded. So in our logic without terms, we need to temporarily replace it with x C_ a and then deduce the original claim. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)

Assertion

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

Proof of Theorem bj-omtrans

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

Step	Hyp	Ref	Expression
1		bj-omex 10067	. . 3 |- om e. _V
2		sseq2 2967	. . . . . 6 |- ( a = om -> ( y C_ a <-> y C_ om ) )
3		sseq2 2967	. . . . . 6 |- ( a = om -> ( suc y C_ a <-> suc y C_ om ) )
4	2, 3	imbi12d 223	. . . . 5 |- ( a = om -> ( ( y C_ a -> suc y C_ a ) <-> ( y C_ om -> suc y C_ om ) ) )
5	4	ralbidv 2326	. . . 4 |- ( a = om -> ( A. y e. om ( y C_ a -> suc y C_ a ) <-> A. y e. om ( y C_ om -> suc y C_ om ) ) )
6		sseq2 2967	. . . . 5 |- ( a = om -> ( A C_ a <-> A C_ om ) )
7	6	imbi2d 219	. . . 4 |- ( a = om -> ( ( A e. om -> A C_ a ) <-> ( A e. om -> A C_ om ) ) )
8	5, 7	imbi12d 223	. . 3 |- ( a = om -> ( ( A. y e. om ( y C_ a -> suc y C_ a ) -> ( A e. om -> A C_ a ) ) <-> ( A. y e. om ( y C_ om -> suc y C_ om ) -> ( A e. om -> A C_ om ) ) ) )
9		0ss 3255	. . . 4 |- (/) C_ a
10		bdcv 9968	. . . . . 6 |- BOUNDED a
11	10	bdss 9984	. . . . 5 |- BOUNDED x C_ a
12		nfv 1421	. . . . 5 |- F/ x (/) C_ a
13		nfv 1421	. . . . 5 |- F/ x y C_ a
14		nfv 1421	. . . . 5 |- F/ x suc y C_ a
15		sseq1 2966	. . . . . 6 |- ( x = (/) -> ( x C_ a <-> (/) C_ a ) )
16	15	biimprd 147	. . . . 5 |- ( x = (/) -> ( (/) C_ a -> x C_ a ) )
17		sseq1 2966	. . . . . 6 |- ( x = y -> ( x C_ a <-> y C_ a ) )
18	17	biimpd 132	. . . . 5 |- ( x = y -> ( x C_ a -> y C_ a ) )
19		sseq1 2966	. . . . . 6 |- ( x = suc y -> ( x C_ a <-> suc y C_ a ) )
20	19	biimprd 147	. . . . 5 |- ( x = suc y -> ( suc y C_ a -> x C_ a ) )
21		nfcv 2178	. . . . 5 |- F/_ x A
22		nfv 1421	. . . . 5 |- F/ x A C_ a
23		sseq1 2966	. . . . . 6 |- ( x = A -> ( x C_ a <-> A C_ a ) )
24	23	biimpd 132	. . . . 5 |- ( x = A -> ( x C_ a -> A C_ a ) )
25	11, 12, 13, 14, 16, 18, 20, 21, 22, 24	bj-bdfindisg 10073	. . . 4 |- ( ( (/) C_ a /\ A. y e. om ( y C_ a -> suc y C_ a ) ) -> ( A e. om -> A C_ a ) )
26	9, 25	mpan 400	. . 3 |- ( A. y e. om ( y C_ a -> suc y C_ a ) -> ( A e. om -> A C_ a ) )
27	1, 8, 26	vtocl 2608	. 2 |- ( A. y e. om ( y C_ om -> suc y C_ om ) -> ( A e. om -> A C_ om ) )
28		df-suc 4108	. . . 4 |- suc y = ( y u. { y } )
29		simpr 103	. . . . 5 |- ( ( y e. om /\ y C_ om ) -> y C_ om )
30		simpl 102	. . . . . 6 |- ( ( y e. om /\ y C_ om ) -> y e. om )
31	30	snssd 3509	. . . . 5 |- ( ( y e. om /\ y C_ om ) -> { y } C_ om )
32	29, 31	unssd 3119	. . . 4 |- ( ( y e. om /\ y C_ om ) -> ( y u. { y } ) C_ om )
33	28, 32	syl5eqss 2989	. . 3 |- ( ( y e. om /\ y C_ om ) -> suc y C_ om )
34	33	ex 108	. 2 |- ( y e. om -> ( y C_ om -> suc y C_ om ) )
35	27, 34	mprg 2378	1 |- ( A e. om -> A C_ om )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   e. wcel 1393   A.wral 2306   u. cun 2915   C_ wss 2917   (/)c0 3224   {csn 3375   suc csuc 4102   omcom 4313

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-nul 3883  ax-pr 3944  ax-un 4170  ax-bd0 9933  ax-bdor 9936  ax-bdal 9938  ax-bdex 9939  ax-bdeq 9940  ax-bdel 9941  ax-bdsb 9942  ax-bdsep 10004  ax-infvn 10066

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-sn 3381  df-pr 3382  df-uni 3581  df-int 3616  df-suc 4108  df-iom 4314  df-bdc 9961  df-bj-ind 10051

This theorem is referenced by:  bj-omtrans2  10082  bj-nnord  10083  bj-nn0suc  10089