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Theorem bj-inf2vn 10486
Description: A sufficient condition for ω to be a set. See bj-inf2vn2 10487 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 10482 . . 3 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → Ind 𝐴)
2 bi1 115 . . . . . . 7 ((𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → (𝑥𝐴 → (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)))
32alimi 1360 . . . . . 6 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → ∀𝑥(𝑥𝐴 → (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)))
4 df-ral 2328 . . . . . 6 (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) ↔ ∀𝑥(𝑥𝐴 → (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)))
53, 4sylibr 141 . . . . 5 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → ∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦))
6 bj-inf2vn.1 . . . . . 6 BOUNDED 𝐴
7 bdcv 10355 . . . . . 6 BOUNDED 𝑧
86, 7bj-inf2vnlem3 10484 . . . . 5 (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑧𝐴𝑧))
95, 8syl 14 . . . 4 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → (Ind 𝑧𝐴𝑧))
109alrimiv 1770 . . 3 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → ∀𝑧(Ind 𝑧𝐴𝑧))
111, 10jca 294 . 2 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → (Ind 𝐴 ∧ ∀𝑧(Ind 𝑧𝐴𝑧)))
12 bj-om 10448 . 2 (𝐴𝑉 → (𝐴 = ω ↔ (Ind 𝐴 ∧ ∀𝑧(Ind 𝑧𝐴𝑧))))
1311, 12syl5ibr 149 1 (𝐴𝑉 → (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → 𝐴 = ω))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wo 639  wal 1257   = wceq 1259  wcel 1409  wral 2323  wrex 2324  wss 2945  c0 3252  suc csuc 4130  ωcom 4341  BOUNDED wbdc 10347  Ind wind 10437
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-nul 3911  ax-pr 3972  ax-un 4198  ax-bd0 10320  ax-bdim 10321  ax-bdor 10323  ax-bdex 10326  ax-bdeq 10327  ax-bdel 10328  ax-bdsb 10329  ax-bdsep 10391  ax-bdsetind 10480
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-sn 3409  df-pr 3410  df-uni 3609  df-int 3644  df-suc 4136  df-iom 4342  df-bdc 10348  df-bj-ind 10438
This theorem is referenced by:  bj-omex2  10489  bj-nn0sucALT  10490
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