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Theorem bj-om 9396
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 9393 . . . 4 Ind 𝜔
2 bj-indeq 9388 . . . 4 (A = 𝜔 → (Ind A ↔ Ind 𝜔))
31, 2mpbiri 157 . . 3 (A = 𝜔 → Ind A)
4 vex 2554 . . . . . 6 x V
5 bj-omssind 9394 . . . . . 6 (x V → (Ind x → 𝜔 ⊆ x))
64, 5ax-mp 7 . . . . 5 (Ind x → 𝜔 ⊆ x)
7 sseq1 2960 . . . . 5 (A = 𝜔 → (Ax ↔ 𝜔 ⊆ x))
86, 7syl5ibr 145 . . . 4 (A = 𝜔 → (Ind xAx))
98alrimiv 1751 . . 3 (A = 𝜔 → x(Ind xAx))
103, 9jca 290 . 2 (A = 𝜔 → (Ind A x(Ind xAx)))
11 bj-ssom 9395 . . . . . . 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 9394 . . . . 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 2954 . . 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 1240   = wceq 1242   wcel 1390  Vcvv 2551  wss 2911  𝜔com 4256  Ind wind 9385
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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-nul 3874  ax-pr 3935  ax-un 4136  ax-bd0 9268  ax-bdor 9271  ax-bdex 9274  ax-bdeq 9275  ax-bdel 9276  ax-bdsb 9277  ax-bdsep 9339
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-sn 3373  df-pr 3374  df-uni 3572  df-int 3607  df-suc 4074  df-iom 4257  df-bdc 9296  df-bj-ind 9386
This theorem is referenced by:  bj-2inf  9397  bj-inf2vn  9434  bj-inf2vn2  9435
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