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Theorem unbnn 8257
Description: Any unbounded subset of natural numbers is equinumerous to the set of all natural numbers. Part of the proof of Theorem 42 of [Suppes] p. 151. See unbnn3 8594 for a stronger version without the first assumption. (Contributed by NM, 3-Dec-2003.)
Assertion
Ref Expression
unbnn ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐴 𝑥𝑦) → 𝐴 ≈ ω)
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem unbnn
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 ssdomg 8043 . . . 4 (ω ∈ V → (𝐴 ⊆ ω → 𝐴 ≼ ω))
21imp 444 . . 3 ((ω ∈ V ∧ 𝐴 ⊆ ω) → 𝐴 ≼ ω)
323adant3 1101 . 2 ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐴 𝑥𝑦) → 𝐴 ≼ ω)
4 simp1 1081 . . 3 ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐴 𝑥𝑦) → ω ∈ V)
5 ssexg 4837 . . . . 5 ((𝐴 ⊆ ω ∧ ω ∈ V) → 𝐴 ∈ V)
65ancoms 468 . . . 4 ((ω ∈ V ∧ 𝐴 ⊆ ω) → 𝐴 ∈ V)
763adant3 1101 . . 3 ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐴 𝑥𝑦) → 𝐴 ∈ V)
8 eqid 2651 . . . . 5 (rec((𝑧 ∈ V ↦ (𝐴 ∖ suc 𝑧)), 𝐴) ↾ ω) = (rec((𝑧 ∈ V ↦ (𝐴 ∖ suc 𝑧)), 𝐴) ↾ ω)
98unblem4 8256 . . . 4 ((𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐴 𝑥𝑦) → (rec((𝑧 ∈ V ↦ (𝐴 ∖ suc 𝑧)), 𝐴) ↾ ω):ω–1-1𝐴)
1093adant1 1099 . . 3 ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐴 𝑥𝑦) → (rec((𝑧 ∈ V ↦ (𝐴 ∖ suc 𝑧)), 𝐴) ↾ ω):ω–1-1𝐴)
11 f1dom2g 8015 . . 3 ((ω ∈ V ∧ 𝐴 ∈ V ∧ (rec((𝑧 ∈ V ↦ (𝐴 ∖ suc 𝑧)), 𝐴) ↾ ω):ω–1-1𝐴) → ω ≼ 𝐴)
124, 7, 10, 11syl3anc 1366 . 2 ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐴 𝑥𝑦) → ω ≼ 𝐴)
13 sbth 8121 . 2 ((𝐴 ≼ ω ∧ ω ≼ 𝐴) → 𝐴 ≈ ω)
143, 12, 13syl2anc 694 1 ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐴 𝑥𝑦) → 𝐴 ≈ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1054  wcel 2030  wral 2941  wrex 2942  Vcvv 3231  cdif 3604  wss 3607   cint 4507   class class class wbr 4685  cmpt 4762  cres 5145  suc csuc 5763  1-1wf1 5923  ωcom 7107  reccrdg 7550  cen 7994  cdom 7995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-en 7998  df-dom 7999
This theorem is referenced by:  unbnn2  8258  isfinite2  8259  unbnn3  8594
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