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Theorem unbnn 8762
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 9110 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 8542 . . . 4 (ω ∈ V → (𝐴 ⊆ ω → 𝐴 ≼ ω))
21imp 410 . . 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 5194 . . . . 5 ((𝐴 ⊆ ω ∧ ω ∈ V) → 𝐴 ∈ V)
65ancoms 462 . . . 4 ((ω ∈ V ∧ 𝐴 ⊆ ω) → 𝐴 ∈ V)
763adant3 1129 . . 3 ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐴 𝑥𝑦) → 𝐴 ∈ V)
8 eqid 2801 . . . . 5 (rec((𝑧 ∈ V ↦ (𝐴 ∖ suc 𝑧)), 𝐴) ↾ ω) = (rec((𝑧 ∈ V ↦ (𝐴 ∖ suc 𝑧)), 𝐴) ↾ ω)
98unblem4 8761 . . . 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 8514 . . 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 8625 . 2 ((𝐴 ≼ ω ∧ ω ≼ 𝐴) → 𝐴 ≈ ω)
143, 12, 13syl2anc 587 1 ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐴 𝑥𝑦) → 𝐴 ≈ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084  wcel 2112  wral 3109  wrex 3110  Vcvv 3444  cdif 3881  wss 3884   cint 4841   class class class wbr 5033  cmpt 5113  cres 5525  suc csuc 6165  1-1wf1 6325  ωcom 7564  reccrdg 8032  cen 8493  cdom 8494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-om 7565  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-en 8497  df-dom 8498
This theorem is referenced by:  unbnn2  8763  isfinite2  8764  unbnn3  9110
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