Theorem bj-om 10061

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 10058	. . . 4 |- Ind om
2		bj-indeq 10053	. . . 4 |- ( A = om -> (Ind A <-> Ind om ) )
3	1, 2	mpbiri 157	. . 3 |- ( A = om -> Ind A )
4		vex 2560	. . . . . 6 |- x e. _V
5		bj-omssind 10059	. . . . . 6 |- ( x e. _V -> (Ind x -> om C_ x ) )
6	4, 5	ax-mp 7	. . . . 5 |- (Ind x -> om C_ x )
7		sseq1 2966	. . . . 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 1754	. . 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 10060	. . . . . . 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 10059	. . . . 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 2960	. . 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 1241   = wceq 1243   e. wcel 1393   _Vcvv 2557   C_ wss 2917   omcom 4313  Ind wind 10050

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-nul 3883  ax-pr 3944  ax-un 4170  ax-bd0 9933  ax-bdor 9936  ax-bdex 9939  ax-bdeq 9940  ax-bdel 9941  ax-bdsb 9942  ax-bdsep 10004

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-sn 3381  df-pr 3382  df-uni 3581  df-int 3616  df-suc 4108  df-iom 4314  df-bdc 9961  df-bj-ind 10051

This theorem is referenced by:  bj-2inf  10062  bj-inf2vn  10099  bj-inf2vn2  10100