Theorem bj-om 9396

Description: A set is equal to om if and only if it is the smallest inductive set. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-om	|- (A e. V -> (A = om <-> (Ind A /\ A.x(Ind x -> A C_ x))))

Distinct variable group:   x,A

Allowed substitution hint:   V(x)

Proof of Theorem bj-om

Step	Hyp	Ref	Expression
1		bj-omind 9393	. . . 4 |- Ind om
2		bj-indeq 9388	. . . 4 |- (A = om -> (Ind A <-> Ind om))
3	1, 2	mpbiri 157	. . 3 |- (A = om -> Ind A)
4		vex 2554	. . . . . 6 |- x e. _V
5		bj-omssind 9394	. . . . . 6 |- (x e. _V -> (Ind x -> om C_ x))
6	4, 5	ax-mp 7	. . . . 5 |- (Ind x -> om C_ x)
7		sseq1 2960	. . . . 5 |- (A = om -> (A C_ x <-> om C_ x))
8	6, 7	syl5ibr 145	. . . 4 |- (A = om -> (Ind x -> A C_ x))
9	8	alrimiv 1751	. . 3 |- (A = om -> A.x(Ind x -> A C_ x))
10	3, 9	jca 290	. 2 |- (A = om -> (Ind A /\ A.x(Ind x -> A C_ x)))
11		bj-ssom 9395	. . . . . . 7 |- (A.x(Ind x -> A C_ x) <-> A C_ om)
12	11	biimpi 113	. . . . . 6 |- (A.x(Ind x -> A C_ x) -> A C_ om)
13	12	adantl 262	. . . . 5 |- ((Ind A /\ A.x(Ind x -> A C_ x)) -> A C_ om)
14	13	a1i 9	. . . 4 |- (A e. V -> ((Ind A /\ A.x(Ind x -> A C_ x)) -> A C_ om))
15		bj-omssind 9394	. . . . 5 |- (A e. V -> (Ind A -> om C_ A))
16	15	adantrd 264	. . . 4 |- (A e. V -> ((Ind A /\ A.x(Ind x -> A C_ x)) -> om C_ A))
17	14, 16	jcad 291	. . 3 |- (A e. V -> ((Ind A /\ A.x(Ind x -> A C_ x)) -> (A C_ om /\ om C_ A)))
18		eqss 2954	. . 3 |- (A = om <-> (A C_ om /\ om C_ A))
19	17, 18	syl6ibr 151	. 2 |- (A e. V -> ((Ind A /\ A.x(Ind x -> A C_ x)) -> A = om))
20	10, 19	impbid2 131	1 |- (A e. V -> (A = om <-> (Ind A /\ A.x(Ind x -> A C_ x))))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98  A.wal 1240   = wceq 1242   e. wcel 1390   _Vcvv 2551   C_ wss 2911   omcom 4256  Ind wind 9385

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-bdex 9274  ax-bdeq 9275  ax-bdel 9276  ax-bdsb 9277  ax-bdsep 9339

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-2inf  9397  bj-inf2vn  9434  bj-inf2vn2  9435