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Theorem bj-inf2vn2 13364
 Description: A sufficient condition for ω to be a set; unbounded version of bj-inf2vn 13363. (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 13359 . . 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 1432 . . . . . 6 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → ∀𝑥(𝑥𝐴 → (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)))
4 df-ral 2422 . . . . . 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 13362 . . . . 5 (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑧𝐴𝑧))
75, 6syl 14 . . . 4 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → (Ind 𝑧𝐴𝑧))
87alrimiv 1847 . . 3 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → ∀𝑧(Ind 𝑧𝐴𝑧))
91, 8jca 304 . 2 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → (Ind 𝐴 ∧ ∀𝑧(Ind 𝑧𝐴𝑧)))
10 bj-om 13326 . 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 698  ∀wal 1330   = wceq 1332   ∈ wcel 1481  ∀wral 2417  ∃wrex 2418   ⊆ wss 3077  ∅c0 3369  suc csuc 4296  ωcom 4513  Ind wind 13315 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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-nul 4063  ax-pr 4140  ax-un 4364  ax-setind 4461  ax-bd0 13202  ax-bdor 13205  ax-bdex 13208  ax-bdeq 13209  ax-bdel 13210  ax-bdsb 13211  ax-bdsep 13273 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2692  df-dif 3079  df-un 3081  df-in 3083  df-ss 3090  df-nul 3370  df-sn 3539  df-pr 3540  df-uni 3746  df-int 3781  df-suc 4302  df-iom 4514  df-bdc 13230  df-bj-ind 13316 This theorem is referenced by: (None)
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