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Theorem bj-om 7306
Description: A set is equal to 𝜔 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 𝑉 → (A = 𝜔 ↔ (Ind A x(Ind xAx))))
Distinct variable group:   x,A
Allowed substitution hint:   𝑉(x)

Proof of Theorem bj-om
StepHypRef Expression
1 bj-omind 7303 . . . 4 Ind 𝜔
2 bj-indeq 7299 . . . 4 (A = 𝜔 → (Ind A ↔ Ind 𝜔))
31, 2mpbiri 157 . . 3 (A = 𝜔 → Ind A)
4 vex 2538 . . . . . 6 x V
5 bj-omssind 7304 . . . . . 6 (x V → (Ind x → 𝜔 ⊆ x))
64, 5ax-mp 7 . . . . 5 (Ind x → 𝜔 ⊆ x)
7 sseq1 2943 . . . . 5 (A = 𝜔 → (Ax ↔ 𝜔 ⊆ x))
86, 7syl5ibr 145 . . . 4 (A = 𝜔 → (Ind xAx))
98alrimiv 1736 . . 3 (A = 𝜔 → x(Ind xAx))
103, 9jca 290 . 2 (A = 𝜔 → (Ind A x(Ind xAx)))
11 bj-ssom 7305 . . . . . . 7 (x(Ind xAx) ↔ A ⊆ 𝜔)
1211biimpi 113 . . . . . 6 (x(Ind xAx) → A ⊆ 𝜔)
1312adantl 262 . . . . 5 ((Ind A x(Ind xAx)) → A ⊆ 𝜔)
1413a1i 9 . . . 4 (A 𝑉 → ((Ind A x(Ind xAx)) → A ⊆ 𝜔))
15 bj-omssind 7304 . . . . 5 (A 𝑉 → (Ind A → 𝜔 ⊆ A))
1615adantrd 264 . . . 4 (A 𝑉 → ((Ind A x(Ind xAx)) → 𝜔 ⊆ A))
1714, 16jcad 291 . . 3 (A 𝑉 → ((Ind A x(Ind xAx)) → (A ⊆ 𝜔 𝜔 ⊆ A)))
18 eqss 2937 . . 3 (A = 𝜔 ↔ (A ⊆ 𝜔 𝜔 ⊆ A))
1917, 18syl6ibr 151 . 2 (A 𝑉 → ((Ind A x(Ind xAx)) → A = 𝜔))
2010, 19impbid2 131 1 (A 𝑉 → (A = 𝜔 ↔ (Ind A x(Ind xAx))))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1226   = wceq 1228   wcel 1374  Vcvv 2535  wss 2894  𝜔com 4240  Ind wind 7296
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-nul 3857  ax-pr 3918  ax-un 4120  ax-bd0 7187  ax-bdor 7190  ax-bdex 7193  ax-bdeq 7194  ax-bdel 7195  ax-bdsb 7196  ax-bdsep 7258
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-rab 2293  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-sn 3356  df-pr 3357  df-uni 3555  df-int 3590  df-suc 4057  df-iom 4241  df-bdc 7215  df-bj-ind 7297
This theorem is referenced by:  bj-2inf  7307  bj-inf2vn  7339  bj-inf2vn2  7340
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