Theorem bj-nntrans 9411

Description: A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)

Assertion

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

Proof of Theorem bj-nntrans

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

Step	Hyp	Ref	Expression
1		ral0 3316	. . 3 |- A.x e. (/) x C_ (/)
2		df-suc 4074	. . . . . . 7 |- suc z = (z u. {z })
3	2	eleq2i 2101	. . . . . 6 |- (x e. suc z <-> x e. (z u. {z }))
4		elun 3078	. . . . . . 7 |- (x e. (z u. {z }) <-> (x e. z \/ x e. {z }))
5		sssucid 4118	. . . . . . . . . 10 |- z C_ suc z
6		sstr2 2946	. . . . . . . . . 10 |- (x C_ z -> (z C_ suc z -> x C_ suc z))
7	5, 6	mpi 15	. . . . . . . . 9 |- (x C_ z -> x C_ suc z)
8	7	imim2i 12	. . . . . . . 8 |- ((x e. z -> x C_ z) -> (x e. z -> x C_ suc z))
9		elsni 3391	. . . . . . . . . 10 |- (x e. {z } -> x = z)
10	9, 5	syl6eqss 2989	. . . . . . . . 9 |- (x e. {z } -> x C_ suc z)
11	10	a1i 9	. . . . . . . 8 |- ((x e. z -> x C_ z) -> (x e. {z } -> x C_ suc z))
12	8, 11	jaod 636	. . . . . . 7 |- ((x e. z -> x C_ z) -> ((x e. z \/ x e. {z }) -> x C_ suc z))
13	4, 12	syl5bi 141	. . . . . 6 |- ((x e. z -> x C_ z) -> (x e. (z u. {z }) -> x C_ suc z))
14	3, 13	syl5bi 141	. . . . 5 |- ((x e. z -> x C_ z) -> (x e. suc z -> x C_ suc z))
15	14	ralimi2 2375	. . . 4 |- (A.x e. z x C_ z -> A.x e. suc zx C_ suc z)
16	15	rgenw 2370	. . 3 |- A.z e. om (A.x e. z x C_ z -> A.x e. suc zx C_ suc z)
17		bdcv 9303	. . . . . 6 |- BOUNDED y
18	17	bdss 9319	. . . . 5 |- BOUNDED x C_ y
19	18	ax-bdal 9273	. . . 4 |- BOUNDED A.x e. y x C_ y
20		nfv 1418	. . . 4 |- F/yA.x e. (/) x C_ (/)
21		nfv 1418	. . . 4 |- F/yA.x e. z x C_ z
22		nfv 1418	. . . 4 |- F/yA.x e. suc zx C_ suc z
23		sseq2 2961	. . . . . 6 |- (y = (/) -> (x C_ y <-> x C_ (/)))
24	23	raleqbi1dv 2507	. . . . 5 |- (y = (/) -> (A.x e. y x C_ y <-> A.x e. (/) x C_ (/)))
25	24	biimprd 147	. . . 4 |- (y = (/) -> (A.x e. (/) x C_ (/) -> A.x e. y x C_ y))
26		sseq2 2961	. . . . . 6 |- (y = z -> (x C_ y <-> x C_ z))
27	26	raleqbi1dv 2507	. . . . 5 |- (y = z -> (A.x e. y x C_ y <-> A.x e. z x C_ z))
28	27	biimpd 132	. . . 4 |- (y = z -> (A.x e. y x C_ y -> A.x e. z x C_ z))
29		sseq2 2961	. . . . . 6 |- (y = suc z -> (x C_ y <-> x C_ suc z))
30	29	raleqbi1dv 2507	. . . . 5 |- (y = suc z -> (A.x e. y x C_ y <-> A.x e. suc zx C_ suc z))
31	30	biimprd 147	. . . 4 |- (y = suc z -> (A.x e. suc zx C_ suc z -> A.x e. y x C_ y))
32		nfcv 2175	. . . 4 |- F/_yA
33		nfv 1418	. . . 4 |- F/yA.x e. A x C_ A
34		sseq2 2961	. . . . . 6 |- (y = A -> (x C_ y <-> x C_ A))
35	34	raleqbi1dv 2507	. . . . 5 |- (y = A -> (A.x e. y x C_ y <-> A.x e. A x C_ A))
36	35	biimpd 132	. . . 4 |- (y = A -> (A.x e. y x C_ y -> A.x e. A x C_ A))
37	19, 20, 21, 22, 25, 28, 31, 32, 33, 36	bj-bdfindisg 9408	. . 3 |- ((A.x e. (/) x C_ (/) /\ A.z e. om (A.x e. z x C_ z -> A.x e. suc zx C_ suc z)) -> (A e. om -> A.x e. A x C_ A))
38	1, 16, 37	mp2an 402	. 2 |- (A e. om -> A.x e. A x C_ A)
39		nfv 1418	. . 3 |- F/x B C_ A
40		sseq1 2960	. . 3 |- (x = B -> (x C_ A <-> B C_ A))
41	39, 40	rspc 2644	. 2 |- (B e. A -> (A.x e. A x C_ A -> B C_ A))
42	38, 41	syl5com 26	1 |- (A e. om -> (B e. A -> B C_ A))

Syntax hints:   -> wi 4   \/ wo 628   = wceq 1242   e. wcel 1390  A.wral 2300   u. cun 2909   C_ wss 2911   (/)c0 3218   {csn 3367   suc csuc 4068   omcom 4256

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-nul 3874  ax-pr 3935  ax-un 4136  ax-bd0 9268  ax-bdor 9271  ax-bdal 9273  ax-bdex 9274  ax-bdeq 9275  ax-bdel 9276  ax-bdsb 9277  ax-bdsep 9339  ax-infvn 9401

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-sn 3373  df-pr 3374  df-uni 3572  df-int 3607  df-suc 4074  df-iom 4257  df-bdc 9296  df-bj-ind 9386

This theorem is referenced by:  bj-nntrans2  9412  bj-nnelirr  9413  bj-nnen2lp  9414