Theorem bj-omtrans 14869

Description: The set om is transitive. A natural number is included in om. Constructive proof of elnn 4607.

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 14855	. . 3 |- om e. _V
2		sseq2 3181	. . . . . 6 |- ( a = om -> ( y C_ a <-> y C_ om ) )
3		sseq2 3181	. . . . . 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 2477	. . . 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 3181	. . . . 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 3463	. . . 4 |- (/) C_ a
10		bdcv 14761	. . . . . 6 |- BOUNDED a
11	10	bdss 14777	. . . . 5 |- BOUNDED x C_ a
12		nfv 1528	. . . . 5 |- F/ x (/) C_ a
13		nfv 1528	. . . . 5 |- F/ x y C_ a
14		nfv 1528	. . . . 5 |- F/ x suc y C_ a
15		sseq1 3180	. . . . . 6 |- ( x = (/) -> ( x C_ a <-> (/) C_ a ) )
16	15	biimprd 158	. . . . 5 |- ( x = (/) -> ( (/) C_ a -> x C_ a ) )
17		sseq1 3180	. . . . . 6 |- ( x = y -> ( x C_ a <-> y C_ a ) )
18	17	biimpd 144	. . . . 5 |- ( x = y -> ( x C_ a -> y C_ a ) )
19		sseq1 3180	. . . . . 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 2319	. . . . 5 |- F/_ x A
22		nfv 1528	. . . . 5 |- F/ x A C_ a
23		sseq1 3180	. . . . . 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 14861	. . . 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 2793	. 2 |- ( A. y e. om ( y C_ om -> suc y C_ om ) -> ( A e. om -> A C_ om ) )
28		df-suc 4373	. . . 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 3739	. . . . 5 |- ( ( y e. om /\ y C_ om ) -> { y } C_ om )
32	29, 31	unssd 3313	. . . 4 |- ( ( y e. om /\ y C_ om ) -> ( y u. { y } ) C_ om )
33	28, 32	eqsstrid 3203	. . 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 2534	1 |- ( A e. om -> A C_ om )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   A.wral 2455   u. cun 3129   C_ wss 3131   (/)c0 3424   {csn 3594   suc csuc 4367   omcom 4591

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-nul 4131  ax-pr 4211  ax-un 4435  ax-bd0 14726  ax-bdor 14729  ax-bdal 14731  ax-bdex 14732  ax-bdeq 14733  ax-bdel 14734  ax-bdsb 14735  ax-bdsep 14797  ax-infvn 14854

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-sn 3600  df-pr 3601  df-uni 3812  df-int 3847  df-suc 4373  df-iom 4592  df-bdc 14754  df-bj-ind 14840

This theorem is referenced by:  bj-omtrans2  14870  bj-nnord  14871  bj-nn0suc  14877