Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bj-inf2vn2 GIF version

Theorem bj-inf2vn2 13007
Description: A sufficient condition for ω to be a set; unbounded version of bj-inf2vn 13006. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-inf2vn2 (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-inf2vn2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 bj-inf2vnlem1 13002 . . 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 1414 . . . . . 6 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → ∀𝑥(𝑥𝐴 → (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)))
4 df-ral 2396 . . . . . 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-inf2vnlem4 13005 . . . . 5 (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑧𝐴𝑧))
75, 6syl 14 . . . 4 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → (Ind 𝑧𝐴𝑧))
87alrimiv 1828 . . 3 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → ∀𝑧(Ind 𝑧𝐴𝑧))
91, 8jca 302 . 2 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → (Ind 𝐴 ∧ ∀𝑧(Ind 𝑧𝐴𝑧)))
10 bj-om 12969 . 2 (𝐴𝑉 → (𝐴 = ω ↔ (Ind 𝐴 ∧ ∀𝑧(Ind 𝑧𝐴𝑧))))
119, 10syl5ibr 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 680  wal 1312   = wceq 1314  wcel 1463  wral 2391  wrex 2392  wss 3039  c0 3331  suc csuc 4255  ωcom 4472  Ind wind 12958
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-setind 4420  ax-bd0 12845  ax-bdor 12848  ax-bdex 12851  ax-bdeq 12852  ax-bdel 12853  ax-bdsb 12854  ax-bdsep 12916
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 12873  df-bj-ind 12959
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator