Theorem bj-omtrans 16091

Description: The set om is transitive. A natural number is included in om. Constructive proof of elnn 4672.

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 16077	. . 3 |- om e. _V
2		sseq2 3225	. . . . . 6 |- ( a = om -> ( y C_ a <-> y C_ om ) )
3		sseq2 3225	. . . . . 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 2508	. . . 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 3225	. . . . 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 3507	. . . 4 |- (/) C_ a
10		bdcv 15983	. . . . . 6 |- BOUNDED a
11	10	bdss 15999	. . . . 5 |- BOUNDED x C_ a
12		nfv 1552	. . . . 5 |- F/ x (/) C_ a
13		nfv 1552	. . . . 5 |- F/ x y C_ a
14		nfv 1552	. . . . 5 |- F/ x suc y C_ a
15		sseq1 3224	. . . . . 6 |- ( x = (/) -> ( x C_ a <-> (/) C_ a ) )
16	15	biimprd 158	. . . . 5 |- ( x = (/) -> ( (/) C_ a -> x C_ a ) )
17		sseq1 3224	. . . . . 6 |- ( x = y -> ( x C_ a <-> y C_ a ) )
18	17	biimpd 144	. . . . 5 |- ( x = y -> ( x C_ a -> y C_ a ) )
19		sseq1 3224	. . . . . 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 2350	. . . . 5 |- F/_ x A
22		nfv 1552	. . . . 5 |- F/ x A C_ a
23		sseq1 3224	. . . . . 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 16083	. . . 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 2832	. 2 |- ( A. y e. om ( y C_ om -> suc y C_ om ) -> ( A e. om -> A C_ om ) )
28		df-suc 4436	. . . 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 3789	. . . . 5 |- ( ( y e. om /\ y C_ om ) -> { y } C_ om )
32	29, 31	unssd 3357	. . . 4 |- ( ( y e. om /\ y C_ om ) -> ( y u. { y } ) C_ om )
33	28, 32	eqsstrid 3247	. . 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 2565	1 |- ( A e. om -> A C_ om )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2178   A.wral 2486   u. cun 3172   C_ wss 3174   (/)c0 3468   {csn 3643   suc csuc 4430   omcom 4656

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-nul 4186  ax-pr 4269  ax-un 4498  ax-bd0 15948  ax-bdor 15951  ax-bdal 15953  ax-bdex 15954  ax-bdeq 15955  ax-bdel 15956  ax-bdsb 15957  ax-bdsep 16019  ax-infvn 16076

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-sn 3649  df-pr 3650  df-uni 3865  df-int 3900  df-suc 4436  df-iom 4657  df-bdc 15976  df-bj-ind 16062

This theorem is referenced by:  bj-omtrans2  16092  bj-nnord  16093  bj-nn0suc  16099