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Theorem bj-inf2vn2 14349
Description: A sufficient condition for ω to be a set; unbounded version of bj-inf2vn 14348. (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 14344 . . 3 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → Ind 𝐴)
2 biimp 118 . . . . . . 7 ((𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → (𝑥𝐴 → (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)))
32alimi 1455 . . . . . 6 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → ∀𝑥(𝑥𝐴 → (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)))
4 df-ral 2460 . . . . . 6 (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) ↔ ∀𝑥(𝑥𝐴 → (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)))
53, 4sylibr 134 . . . . 5 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → ∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦))
6 bj-inf2vnlem4 14347 . . . . 5 (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑧𝐴𝑧))
75, 6syl 14 . . . 4 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → (Ind 𝑧𝐴𝑧))
87alrimiv 1874 . . 3 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → ∀𝑧(Ind 𝑧𝐴𝑧))
91, 8jca 306 . 2 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → (Ind 𝐴 ∧ ∀𝑧(Ind 𝑧𝐴𝑧)))
10 bj-om 14311 . 2 (𝐴𝑉 → (𝐴 = ω ↔ (Ind 𝐴 ∧ ∀𝑧(Ind 𝑧𝐴𝑧))))
119, 10syl5ibr 156 1 (𝐴𝑉 → (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → 𝐴 = ω))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wo 708  wal 1351   = wceq 1353  wcel 2148  wral 2455  wrex 2456  wss 3129  c0 3422  suc csuc 4361  ωcom 4585  Ind wind 14300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-nul 4126  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-bd0 14187  ax-bdor 14190  ax-bdex 14193  ax-bdeq 14194  ax-bdel 14195  ax-bdsb 14196  ax-bdsep 14258
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-sn 3597  df-pr 3598  df-uni 3808  df-int 3843  df-suc 4367  df-iom 4586  df-bdc 14215  df-bj-ind 14301
This theorem is referenced by: (None)
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