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Theorem bj-om 12958
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 12955 . . . 4 Ind ω
2 bj-indeq 12950 . . . 4 (𝐴 = ω → (Ind 𝐴 ↔ Ind ω))
31, 2mpbiri 167 . . 3 (𝐴 = ω → Ind 𝐴)
4 vex 2661 . . . . . 6 𝑥 ∈ V
5 bj-omssind 12956 . . . . . 6 (𝑥 ∈ V → (Ind 𝑥 → ω ⊆ 𝑥))
64, 5ax-mp 5 . . . . 5 (Ind 𝑥 → ω ⊆ 𝑥)
7 sseq1 3088 . . . . 5 (𝐴 = ω → (𝐴𝑥 ↔ ω ⊆ 𝑥))
86, 7syl5ibr 155 . . . 4 (𝐴 = ω → (Ind 𝑥𝐴𝑥))
98alrimiv 1828 . . 3 (𝐴 = ω → ∀𝑥(Ind 𝑥𝐴𝑥))
103, 9jca 302 . 2 (𝐴 = ω → (Ind 𝐴 ∧ ∀𝑥(Ind 𝑥𝐴𝑥)))
11 bj-ssom 12957 . . . . . . 7 (∀𝑥(Ind 𝑥𝐴𝑥) ↔ 𝐴 ⊆ ω)
1211biimpi 119 . . . . . 6 (∀𝑥(Ind 𝑥𝐴𝑥) → 𝐴 ⊆ ω)
1312adantl 273 . . . . 5 ((Ind 𝐴 ∧ ∀𝑥(Ind 𝑥𝐴𝑥)) → 𝐴 ⊆ ω)
1413a1i 9 . . . 4 (𝐴𝑉 → ((Ind 𝐴 ∧ ∀𝑥(Ind 𝑥𝐴𝑥)) → 𝐴 ⊆ ω))
15 bj-omssind 12956 . . . . 5 (𝐴𝑉 → (Ind 𝐴 → ω ⊆ 𝐴))
1615adantrd 275 . . . 4 (𝐴𝑉 → ((Ind 𝐴 ∧ ∀𝑥(Ind 𝑥𝐴𝑥)) → ω ⊆ 𝐴))
1714, 16jcad 303 . . 3 (𝐴𝑉 → ((Ind 𝐴 ∧ ∀𝑥(Ind 𝑥𝐴𝑥)) → (𝐴 ⊆ ω ∧ ω ⊆ 𝐴)))
18 eqss 3080 . . 3 (𝐴 = ω ↔ (𝐴 ⊆ ω ∧ ω ⊆ 𝐴))
1917, 18syl6ibr 161 . 2 (𝐴𝑉 → ((Ind 𝐴 ∧ ∀𝑥(Ind 𝑥𝐴𝑥)) → 𝐴 = ω))
2010, 19impbid2 142 1 (𝐴𝑉 → (𝐴 = ω ↔ (Ind 𝐴 ∧ ∀𝑥(Ind 𝑥𝐴𝑥))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1312   = wceq 1314  wcel 1463  Vcvv 2658  wss 3039  ωcom 4472  Ind wind 12947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-nul 4022  ax-pr 4099  ax-un 4323  ax-bd0 12834  ax-bdor 12837  ax-bdex 12840  ax-bdeq 12841  ax-bdel 12842  ax-bdsb 12843  ax-bdsep 12905
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-sn 3501  df-pr 3502  df-uni 3705  df-int 3740  df-suc 4261  df-iom 4473  df-bdc 12862  df-bj-ind 12948
This theorem is referenced by:  bj-2inf  12959  bj-inf2vn  12995  bj-inf2vn2  12996
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