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Theorem bj-inf2vn 12757
Description: A sufficient condition for ω to be a set. See bj-inf2vn2 12758 for the unbounded version from full set induction. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-inf2vn.1 BOUNDED 𝐴
Assertion
Ref Expression
bj-inf2vn (𝐴𝑉 → (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → 𝐴 = ω))
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝑉(𝑥,𝑦)

Proof of Theorem bj-inf2vn
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 bj-inf2vnlem1 12753 . . 3 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → Ind 𝐴)
2 bi1 117 . . . . . . 7 ((𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → (𝑥𝐴 → (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)))
32alimi 1399 . . . . . 6 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → ∀𝑥(𝑥𝐴 → (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)))
4 df-ral 2380 . . . . . 6 (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) ↔ ∀𝑥(𝑥𝐴 → (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)))
53, 4sylibr 133 . . . . 5 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → ∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦))
6 bj-inf2vn.1 . . . . . 6 BOUNDED 𝐴
7 bdcv 12627 . . . . . 6 BOUNDED 𝑧
86, 7bj-inf2vnlem3 12755 . . . . 5 (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑧𝐴𝑧))
95, 8syl 14 . . . 4 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → (Ind 𝑧𝐴𝑧))
109alrimiv 1813 . . 3 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → ∀𝑧(Ind 𝑧𝐴𝑧))
111, 10jca 302 . 2 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → (Ind 𝐴 ∧ ∀𝑧(Ind 𝑧𝐴𝑧)))
12 bj-om 12720 . 2 (𝐴𝑉 → (𝐴 = ω ↔ (Ind 𝐴 ∧ ∀𝑧(Ind 𝑧𝐴𝑧))))
1311, 12syl5ibr 155 1 (𝐴𝑉 → (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → 𝐴 = ω))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wo 670  wal 1297   = wceq 1299  wcel 1448  wral 2375  wrex 2376  wss 3021  c0 3310  suc csuc 4225  ωcom 4442  BOUNDED wbdc 12619  Ind wind 12709
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-nul 3994  ax-pr 4069  ax-un 4293  ax-bd0 12592  ax-bdim 12593  ax-bdor 12595  ax-bdex 12598  ax-bdeq 12599  ax-bdel 12600  ax-bdsb 12601  ax-bdsep 12663  ax-bdsetind 12751
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-sn 3480  df-pr 3481  df-uni 3684  df-int 3719  df-suc 4231  df-iom 4443  df-bdc 12620  df-bj-ind 12710
This theorem is referenced by:  bj-omex2  12760  bj-nn0sucALT  12761
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