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Theorem unbnn 9296
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 9651 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 8993 . . . 4 (ω ∈ V → (𝐴 ⊆ ω → 𝐴 ≼ ω))
21imp 406 . . 3 ((ω ∈ V ∧ 𝐴 ⊆ ω) → 𝐴 ≼ ω)
323adant3 1129 . 2 ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐴 𝑥𝑦) → 𝐴 ≼ ω)
4 simp1 1133 . . 3 ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐴 𝑥𝑦) → ω ∈ V)
5 ssexg 5314 . . . . 5 ((𝐴 ⊆ ω ∧ ω ∈ V) → 𝐴 ∈ V)
65ancoms 458 . . . 4 ((ω ∈ V ∧ 𝐴 ⊆ ω) → 𝐴 ∈ V)
763adant3 1129 . . 3 ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐴 𝑥𝑦) → 𝐴 ∈ V)
8 eqid 2724 . . . . 5 (rec((𝑧 ∈ V ↦ (𝐴 ∖ suc 𝑧)), 𝐴) ↾ ω) = (rec((𝑧 ∈ V ↦ (𝐴 ∖ suc 𝑧)), 𝐴) ↾ ω)
98unblem4 9295 . . . 4 ((𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐴 𝑥𝑦) → (rec((𝑧 ∈ V ↦ (𝐴 ∖ suc 𝑧)), 𝐴) ↾ ω):ω–1-1𝐴)
1093adant1 1127 . . 3 ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐴 𝑥𝑦) → (rec((𝑧 ∈ V ↦ (𝐴 ∖ suc 𝑧)), 𝐴) ↾ ω):ω–1-1𝐴)
11 f1dom2g 8962 . . 3 ((ω ∈ V ∧ 𝐴 ∈ V ∧ (rec((𝑧 ∈ V ↦ (𝐴 ∖ suc 𝑧)), 𝐴) ↾ ω):ω–1-1𝐴) → ω ≼ 𝐴)
124, 7, 10, 11syl3anc 1368 . 2 ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐴 𝑥𝑦) → ω ≼ 𝐴)
13 sbth 9090 . 2 ((𝐴 ≼ ω ∧ ω ≼ 𝐴) → 𝐴 ≈ ω)
143, 12, 13syl2anc 583 1 ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐴 𝑥𝑦) → 𝐴 ≈ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084  wcel 2098  wral 3053  wrex 3062  Vcvv 3466  cdif 3938  wss 3941   cint 4941   class class class wbr 5139  cmpt 5222  cres 5669  suc csuc 6357  1-1wf1 6531  ωcom 7849  reccrdg 8405  cen 8933  cdom 8934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4942  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-om 7850  df-2nd 7970  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-en 8937  df-dom 8938
This theorem is referenced by:  unbnn2  9297  isfinite2  9298  unbnn3  9651
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