Theorem bj-om 13972

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 13969	. . . 4 |- Ind om
2		bj-indeq 13964	. . . 4 |- ( A = om -> (Ind A <-> Ind om ) )
3	1, 2	mpbiri 167	. . 3 |- ( A = om -> Ind A )
4		vex 2733	. . . . . 6 |- x e. _V
5		bj-omssind 13970	. . . . . 6 |- ( x e. _V -> (Ind x -> om C_ x ) )
6	4, 5	ax-mp 5	. . . . 5 |- (Ind x -> om C_ x )
7		sseq1 3170	. . . . 5 |- ( A = om -> ( A C_ x <-> om C_ x ) )
8	6, 7	syl5ibr 155	. . . 4 |- ( A = om -> (Ind x -> A C_ x ) )
9	8	alrimiv 1867	. . 3 |- ( A = om -> A. x (Ind x -> A C_ x ) )
10	3, 9	jca 304	. 2 |- ( A = om -> (Ind A /\ A. x (Ind x -> A C_ x ) ) )
11		bj-ssom 13971	. . . . . . 7 |- ( A. x (Ind x -> A C_ x ) <-> A C_ om )
12	11	biimpi 119	. . . . . 6 |- ( A. x (Ind x -> A C_ x ) -> A C_ om )
13	12	adantl 275	. . . . 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 13970	. . . . 5 |- ( A e. V -> (Ind A -> om C_ A ) )
16	15	adantrd 277	. . . 4 |- ( A e. V -> ( (Ind A /\ A. x (Ind x -> A C_ x ) ) -> om C_ A ) )
17	14, 16	jcad 305	. . 3 |- ( A e. V -> ( (Ind A /\ A. x (Ind x -> A C_ x ) ) -> ( A C_ om /\ om C_ A ) ) )
18		eqss 3162	. . 3 |- ( A = om <-> ( A C_ om /\ om C_ A ) )
19	17, 18	syl6ibr 161	. 2 |- ( A e. V -> ( (Ind A /\ A. x (Ind x -> A C_ x ) ) -> A = om ) )
20	10, 19	impbid2 142	1 |- ( A e. V -> ( A = om <-> (Ind A /\ A. x (Ind x -> A C_ x ) ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1346   = wceq 1348   e. wcel 2141   _Vcvv 2730   C_ wss 3121   omcom 4574  Ind wind 13961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-nul 4115  ax-pr 4194  ax-un 4418  ax-bd0 13848  ax-bdor 13851  ax-bdex 13854  ax-bdeq 13855  ax-bdel 13856  ax-bdsb 13857  ax-bdsep 13919

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-sn 3589  df-pr 3590  df-uni 3797  df-int 3832  df-suc 4356  df-iom 4575  df-bdc 13876  df-bj-ind 13962

This theorem is referenced by:  bj-2inf  13973  bj-inf2vn  14009  bj-inf2vn2  14010