Theorem bj-om 10917

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 10914	. . . 4 |- Ind om
2		bj-indeq 10909	. . . 4 |- ( A = om -> (Ind A <-> Ind om ) )
3	1, 2	mpbiri 166	. . 3 |- ( A = om -> Ind A )
4		vex 2605	. . . . . 6 |- x e. _V
5		bj-omssind 10915	. . . . . 6 |- ( x e. _V -> (Ind x -> om C_ x ) )
6	4, 5	ax-mp 7	. . . . 5 |- (Ind x -> om C_ x )
7		sseq1 3021	. . . . 5 |- ( A = om -> ( A C_ x <-> om C_ x ) )
8	6, 7	syl5ibr 154	. . . 4 |- ( A = om -> (Ind x -> A C_ x ) )
9	8	alrimiv 1796	. . 3 |- ( A = om -> A. x (Ind x -> A C_ x ) )
10	3, 9	jca 300	. 2 |- ( A = om -> (Ind A /\ A. x (Ind x -> A C_ x ) ) )
11		bj-ssom 10916	. . . . . . 7 |- ( A. x (Ind x -> A C_ x ) <-> A C_ om )
12	11	biimpi 118	. . . . . 6 |- ( A. x (Ind x -> A C_ x ) -> A C_ om )
13	12	adantl 271	. . . . 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 10915	. . . . 5 |- ( A e. V -> (Ind A -> om C_ A ) )
16	15	adantrd 273	. . . 4 |- ( A e. V -> ( (Ind A /\ A. x (Ind x -> A C_ x ) ) -> om C_ A ) )
17	14, 16	jcad 301	. . 3 |- ( A e. V -> ( (Ind A /\ A. x (Ind x -> A C_ x ) ) -> ( A C_ om /\ om C_ A ) ) )
18		eqss 3015	. . 3 |- ( A = om <-> ( A C_ om /\ om C_ A ) )
19	17, 18	syl6ibr 160	. 2 |- ( A e. V -> ( (Ind A /\ A. x (Ind x -> A C_ x ) ) -> A = om ) )
20	10, 19	impbid2 141	1 |- ( A e. V -> ( A = om <-> (Ind A /\ A. x (Ind x -> A C_ x ) ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1283   = wceq 1285   e. wcel 1434   _Vcvv 2602   C_ wss 2974   omcom 4339  Ind wind 10906

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-nul 3912  ax-pr 3972  ax-un 4196  ax-bd0 10789  ax-bdor 10792  ax-bdex 10795  ax-bdeq 10796  ax-bdel 10797  ax-bdsb 10798  ax-bdsep 10860

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-sn 3412  df-pr 3413  df-uni 3610  df-int 3645  df-suc 4134  df-iom 4340  df-bdc 10817  df-bj-ind 10907

This theorem is referenced by:  bj-2inf  10918  bj-inf2vn  10954  bj-inf2vn2  10955