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Theorem bj-om 10061
 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 (𝐴𝑉 → (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 10058 . . . 4 Ind ω
2 bj-indeq 10053 . . . 4 (𝐴 = ω → (Ind 𝐴 ↔ Ind ω))
31, 2mpbiri 157 . . 3 (𝐴 = ω → Ind 𝐴)
4 vex 2560 . . . . . 6 𝑥 ∈ V
5 bj-omssind 10059 . . . . . 6 (𝑥 ∈ V → (Ind 𝑥 → ω ⊆ 𝑥))
64, 5ax-mp 7 . . . . 5 (Ind 𝑥 → ω ⊆ 𝑥)
7 sseq1 2966 . . . . 5 (𝐴 = ω → (𝐴𝑥 ↔ ω ⊆ 𝑥))
86, 7syl5ibr 145 . . . 4 (𝐴 = ω → (Ind 𝑥𝐴𝑥))
98alrimiv 1754 . . 3 (𝐴 = ω → ∀𝑥(Ind 𝑥𝐴𝑥))
103, 9jca 290 . 2 (𝐴 = ω → (Ind 𝐴 ∧ ∀𝑥(Ind 𝑥𝐴𝑥)))
11 bj-ssom 10060 . . . . . . 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 (𝐴𝑉 → ((Ind 𝐴 ∧ ∀𝑥(Ind 𝑥𝐴𝑥)) → 𝐴 ⊆ ω))
15 bj-omssind 10059 . . . . 5 (𝐴𝑉 → (Ind 𝐴 → ω ⊆ 𝐴))
1615adantrd 264 . . . 4 (𝐴𝑉 → ((Ind 𝐴 ∧ ∀𝑥(Ind 𝑥𝐴𝑥)) → ω ⊆ 𝐴))
1714, 16jcad 291 . . 3 (𝐴𝑉 → ((Ind 𝐴 ∧ ∀𝑥(Ind 𝑥𝐴𝑥)) → (𝐴 ⊆ ω ∧ ω ⊆ 𝐴)))
18 eqss 2960 . . 3 (𝐴 = ω ↔ (𝐴 ⊆ ω ∧ ω ⊆ 𝐴))
1917, 18syl6ibr 151 . 2 (𝐴𝑉 → ((Ind 𝐴 ∧ ∀𝑥(Ind 𝑥𝐴𝑥)) → 𝐴 = ω))
2010, 19impbid2 131 1 (𝐴𝑉 → (𝐴 = ω ↔ (Ind 𝐴 ∧ ∀𝑥(Ind 𝑥𝐴𝑥))))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1241   = wceq 1243   ∈ wcel 1393  Vcvv 2557   ⊆ wss 2917  ωcom 4313  Ind wind 10050 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-nul 3883  ax-pr 3944  ax-un 4170  ax-bd0 9933  ax-bdor 9936  ax-bdex 9939  ax-bdeq 9940  ax-bdel 9941  ax-bdsb 9942  ax-bdsep 10004 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-sn 3381  df-pr 3382  df-uni 3581  df-int 3616  df-suc 4108  df-iom 4314  df-bdc 9961  df-bj-ind 10051 This theorem is referenced by:  bj-2inf  10062  bj-inf2vn  10099  bj-inf2vn2  10100
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