Theorem bj-omtrans 16551

Description: The set om is transitive. A natural number is included in om. Constructive proof of elnn 4704.

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 16537	. . 3 |- om e. _V
2		sseq2 3251	. . . . . 6 |- ( a = om -> ( y C_ a <-> y C_ om ) )
3		sseq2 3251	. . . . . 6 |- ( a = om -> ( suc y C_ a <-> suc y C_ om ) )
4	2, 3	imbi12d 234	. . . . 5 |- ( a = om -> ( ( y C_ a -> suc y C_ a ) <-> ( y C_ om -> suc y C_ om ) ) )
5	4	ralbidv 2532	. . . 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 3251	. . . . 5 |- ( a = om -> ( A C_ a <-> A C_ om ) )
7	6	imbi2d 230	. . . 4 |- ( a = om -> ( ( A e. om -> A C_ a ) <-> ( A e. om -> A C_ om ) ) )
8	5, 7	imbi12d 234	. . 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 3533	. . . 4 |- (/) C_ a
10		bdcv 16443	. . . . . 6 |- BOUNDED a
11	10	bdss 16459	. . . . 5 |- BOUNDED x C_ a
12		nfv 1576	. . . . 5 |- F/ x (/) C_ a
13		nfv 1576	. . . . 5 |- F/ x y C_ a
14		nfv 1576	. . . . 5 |- F/ x suc y C_ a
15		sseq1 3250	. . . . . 6 |- ( x = (/) -> ( x C_ a <-> (/) C_ a ) )
16	15	biimprd 158	. . . . 5 |- ( x = (/) -> ( (/) C_ a -> x C_ a ) )
17		sseq1 3250	. . . . . 6 |- ( x = y -> ( x C_ a <-> y C_ a ) )
18	17	biimpd 144	. . . . 5 |- ( x = y -> ( x C_ a -> y C_ a ) )
19		sseq1 3250	. . . . . 6 |- ( x = suc y -> ( x C_ a <-> suc y C_ a ) )
20	19	biimprd 158	. . . . 5 |- ( x = suc y -> ( suc y C_ a -> x C_ a ) )
21		nfcv 2374	. . . . 5 |- F/_ x A
22		nfv 1576	. . . . 5 |- F/ x A C_ a
23		sseq1 3250	. . . . . 6 |- ( x = A -> ( x C_ a <-> A C_ a ) )
24	23	biimpd 144	. . . . 5 |- ( x = A -> ( x C_ a -> A C_ a ) )
25	11, 12, 13, 14, 16, 18, 20, 21, 22, 24	bj-bdfindisg 16543	. . . 4 |- ( ( (/) C_ a /\ A. y e. om ( y C_ a -> suc y C_ a ) ) -> ( A e. om -> A C_ a ) )
26	9, 25	mpan 424	. . 3 |- ( A. y e. om ( y C_ a -> suc y C_ a ) -> ( A e. om -> A C_ a ) )
27	1, 8, 26	vtocl 2858	. 2 |- ( A. y e. om ( y C_ om -> suc y C_ om ) -> ( A e. om -> A C_ om ) )
28		df-suc 4468	. . . 4 |- suc y = ( y u. { y } )
29		simpr 110	. . . . 5 |- ( ( y e. om /\ y C_ om ) -> y C_ om )
30		simpl 109	. . . . . 6 |- ( ( y e. om /\ y C_ om ) -> y e. om )
31	30	snssd 3818	. . . . 5 |- ( ( y e. om /\ y C_ om ) -> { y } C_ om )
32	29, 31	unssd 3383	. . . 4 |- ( ( y e. om /\ y C_ om ) -> ( y u. { y } ) C_ om )
33	28, 32	eqsstrid 3273	. . 3 |- ( ( y e. om /\ y C_ om ) -> suc y C_ om )
34	33	ex 115	. 2 |- ( y e. om -> ( y C_ om -> suc y C_ om ) )
35	27, 34	mprg 2589	1 |- ( A e. om -> A C_ om )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   A.wral 2510   u. cun 3198   C_ wss 3200   (/)c0 3494   {csn 3669   suc csuc 4462   omcom 4688

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-nul 4215  ax-pr 4299  ax-un 4530  ax-bd0 16408  ax-bdor 16411  ax-bdal 16413  ax-bdex 16414  ax-bdeq 16415  ax-bdel 16416  ax-bdsb 16417  ax-bdsep 16479  ax-infvn 16536

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-sn 3675  df-pr 3676  df-uni 3894  df-int 3929  df-suc 4468  df-iom 4689  df-bdc 16436  df-bj-ind 16522

This theorem is referenced by:  bj-omtrans2  16552  bj-nnord  16553  bj-nn0suc  16559