Theorem bj-omtrans 13838

Description: The set om is transitive. A natural number is included in om. Constructive proof of elnn 4583.

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 13824	. . 3 |- om e. _V
2		sseq2 3166	. . . . . 6 |- ( a = om -> ( y C_ a <-> y C_ om ) )
3		sseq2 3166	. . . . . 6 |- ( a = om -> ( suc y C_ a <-> suc y C_ om ) )
4	2, 3	imbi12d 233	. . . . 5 |- ( a = om -> ( ( y C_ a -> suc y C_ a ) <-> ( y C_ om -> suc y C_ om ) ) )
5	4	ralbidv 2466	. . . 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 3166	. . . . 5 |- ( a = om -> ( A C_ a <-> A C_ om ) )
7	6	imbi2d 229	. . . 4 |- ( a = om -> ( ( A e. om -> A C_ a ) <-> ( A e. om -> A C_ om ) ) )
8	5, 7	imbi12d 233	. . 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 3447	. . . 4 |- (/) C_ a
10		bdcv 13730	. . . . . 6 |- BOUNDED a
11	10	bdss 13746	. . . . 5 |- BOUNDED x C_ a
12		nfv 1516	. . . . 5 |- F/ x (/) C_ a
13		nfv 1516	. . . . 5 |- F/ x y C_ a
14		nfv 1516	. . . . 5 |- F/ x suc y C_ a
15		sseq1 3165	. . . . . 6 |- ( x = (/) -> ( x C_ a <-> (/) C_ a ) )
16	15	biimprd 157	. . . . 5 |- ( x = (/) -> ( (/) C_ a -> x C_ a ) )
17		sseq1 3165	. . . . . 6 |- ( x = y -> ( x C_ a <-> y C_ a ) )
18	17	biimpd 143	. . . . 5 |- ( x = y -> ( x C_ a -> y C_ a ) )
19		sseq1 3165	. . . . . 6 |- ( x = suc y -> ( x C_ a <-> suc y C_ a ) )
20	19	biimprd 157	. . . . 5 |- ( x = suc y -> ( suc y C_ a -> x C_ a ) )
21		nfcv 2308	. . . . 5 |- F/_ x A
22		nfv 1516	. . . . 5 |- F/ x A C_ a
23		sseq1 3165	. . . . . 6 |- ( x = A -> ( x C_ a <-> A C_ a ) )
24	23	biimpd 143	. . . . 5 |- ( x = A -> ( x C_ a -> A C_ a ) )
25	11, 12, 13, 14, 16, 18, 20, 21, 22, 24	bj-bdfindisg 13830	. . . 4 |- ( ( (/) C_ a /\ A. y e. om ( y C_ a -> suc y C_ a ) ) -> ( A e. om -> A C_ a ) )
26	9, 25	mpan 421	. . 3 |- ( A. y e. om ( y C_ a -> suc y C_ a ) -> ( A e. om -> A C_ a ) )
27	1, 8, 26	vtocl 2780	. 2 |- ( A. y e. om ( y C_ om -> suc y C_ om ) -> ( A e. om -> A C_ om ) )
28		df-suc 4349	. . . 4 |- suc y = ( y u. { y } )
29		simpr 109	. . . . 5 |- ( ( y e. om /\ y C_ om ) -> y C_ om )
30		simpl 108	. . . . . 6 |- ( ( y e. om /\ y C_ om ) -> y e. om )
31	30	snssd 3718	. . . . 5 |- ( ( y e. om /\ y C_ om ) -> { y } C_ om )
32	29, 31	unssd 3298	. . . 4 |- ( ( y e. om /\ y C_ om ) -> ( y u. { y } ) C_ om )
33	28, 32	eqsstrid 3188	. . 3 |- ( ( y e. om /\ y C_ om ) -> suc y C_ om )
34	33	ex 114	. 2 |- ( y e. om -> ( y C_ om -> suc y C_ om ) )
35	27, 34	mprg 2523	1 |- ( A e. om -> A C_ om )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   A.wral 2444   u. cun 3114   C_ wss 3116   (/)c0 3409   {csn 3576   suc csuc 4343   omcom 4567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-nul 4108  ax-pr 4187  ax-un 4411  ax-bd0 13695  ax-bdor 13698  ax-bdal 13700  ax-bdex 13701  ax-bdeq 13702  ax-bdel 13703  ax-bdsb 13704  ax-bdsep 13766  ax-infvn 13823

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-sn 3582  df-pr 3583  df-uni 3790  df-int 3825  df-suc 4349  df-iom 4568  df-bdc 13723  df-bj-ind 13809

This theorem is referenced by:  bj-omtrans2  13839  bj-nnord  13840  bj-nn0suc  13846