Theorem bj-om 16532

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 16529	. . . 4 |- Ind om
2		bj-indeq 16524	. . . 4 |- ( A = om -> (Ind A <-> Ind om ) )
3	1, 2	mpbiri 168	. . 3 |- ( A = om -> Ind A )
4		vex 2805	. . . . . 6 |- x e. _V
5		bj-omssind 16530	. . . . . 6 |- ( x e. _V -> (Ind x -> om C_ x ) )
6	4, 5	ax-mp 5	. . . . 5 |- (Ind x -> om C_ x )
7		sseq1 3250	. . . . 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 1922	. . 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 16531	. . . . . . 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 16530	. . . . 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 3242	. . 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 1395   = wceq 1397   e. wcel 2202   _Vcvv 2802   C_ wss 3200   omcom 4688  Ind wind 16521

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-nul 4215  ax-pr 4299  ax-un 4530  ax-bd0 16408  ax-bdor 16411  ax-bdex 16414  ax-bdeq 16415  ax-bdel 16416  ax-bdsb 16417  ax-bdsep 16479

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-sn 3675  df-pr 3676  df-uni 3894  df-int 3929  df-suc 4468  df-iom 4689  df-bdc 16436  df-bj-ind 16522

This theorem is referenced by:  bj-2inf  16533  bj-inf2vn  16569  bj-inf2vn2  16570