Theorem bj-om 16300

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 16297	. . . 4 |- Ind om
2		bj-indeq 16292	. . . 4 |- ( A = om -> (Ind A <-> Ind om ) )
3	1, 2	mpbiri 168	. . 3 |- ( A = om -> Ind A )
4		vex 2802	. . . . . 6 |- x e. _V
5		bj-omssind 16298	. . . . . 6 |- ( x e. _V -> (Ind x -> om C_ x ) )
6	4, 5	ax-mp 5	. . . . 5 |- (Ind x -> om C_ x )
7		sseq1 3247	. . . . 5 |- ( A = om -> ( A C_ x <-> om C_ x ) )
8	6, 7	imbitrrid 156	. . . 4 |- ( A = om -> (Ind x -> A C_ x ) )
9	8	alrimiv 1920	. . 3 |- ( A = om -> A. x (Ind x -> A C_ x ) )
10	3, 9	jca 306	. 2 |- ( A = om -> (Ind A /\ A. x (Ind x -> A C_ x ) ) )
11		bj-ssom 16299	. . . . . . 7 |- ( A. x (Ind x -> A C_ x ) <-> A C_ om )
12	11	biimpi 120	. . . . . 6 |- ( A. x (Ind x -> A C_ x ) -> A C_ om )
13	12	adantl 277	. . . . 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 16298	. . . . 5 |- ( A e. V -> (Ind A -> om C_ A ) )
16	15	adantrd 279	. . . 4 |- ( A e. V -> ( (Ind A /\ A. x (Ind x -> A C_ x ) ) -> om C_ A ) )
17	14, 16	jcad 307	. . 3 |- ( A e. V -> ( (Ind A /\ A. x (Ind x -> A C_ x ) ) -> ( A C_ om /\ om C_ A ) ) )
18		eqss 3239	. . 3 |- ( A = om <-> ( A C_ om /\ om C_ A ) )
19	17, 18	imbitrrdi 162	. 2 |- ( A e. V -> ( (Ind A /\ A. x (Ind x -> A C_ x ) ) -> A = om ) )
20	10, 19	impbid2 143	1 |- ( A e. V -> ( A = om <-> (Ind A /\ A. x (Ind x -> A C_ x ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1393   = wceq 1395   e. wcel 2200   _Vcvv 2799   C_ wss 3197   omcom 4682  Ind wind 16289

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-nul 4210  ax-pr 4293  ax-un 4524  ax-bd0 16176  ax-bdor 16179  ax-bdex 16182  ax-bdeq 16183  ax-bdel 16184  ax-bdsb 16185  ax-bdsep 16247

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-sn 3672  df-pr 3673  df-uni 3889  df-int 3924  df-suc 4462  df-iom 4683  df-bdc 16204  df-bj-ind 16290

This theorem is referenced by:  bj-2inf  16301  bj-inf2vn  16337  bj-inf2vn2  16338