Theorem bj-om 15910

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 15907	. . . 4 |- Ind om
2		bj-indeq 15902	. . . 4 |- ( A = om -> (Ind A <-> Ind om ) )
3	1, 2	mpbiri 168	. . 3 |- ( A = om -> Ind A )
4		vex 2775	. . . . . 6 |- x e. _V
5		bj-omssind 15908	. . . . . 6 |- ( x e. _V -> (Ind x -> om C_ x ) )
6	4, 5	ax-mp 5	. . . . 5 |- (Ind x -> om C_ x )
7		sseq1 3216	. . . . 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 1897	. . 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 15909	. . . . . . 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 15908	. . . . 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 3208	. . 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 1371   = wceq 1373   e. wcel 2176   _Vcvv 2772   C_ wss 3166   omcom 4639  Ind wind 15899

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-nul 4171  ax-pr 4254  ax-un 4481  ax-bd0 15786  ax-bdor 15789  ax-bdex 15792  ax-bdeq 15793  ax-bdel 15794  ax-bdsb 15795  ax-bdsep 15857

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-sn 3639  df-pr 3640  df-uni 3851  df-int 3886  df-suc 4419  df-iom 4640  df-bdc 15814  df-bj-ind 15900

This theorem is referenced by:  bj-2inf  15911  bj-inf2vn  15947  bj-inf2vn2  15948