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Theorem unbnn 9295
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 9650 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 8992 . . . 4 (ω ∈ V → (𝐴 ⊆ ω → 𝐴 ≼ ω))
21imp 407 . . 3 ((ω ∈ V ∧ 𝐴 ⊆ ω) → 𝐴 ≼ ω)
323adant3 1132 . 2 ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐴 𝑥𝑦) → 𝐴 ≼ ω)
4 simp1 1136 . . 3 ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐴 𝑥𝑦) → ω ∈ V)
5 ssexg 5322 . . . . 5 ((𝐴 ⊆ ω ∧ ω ∈ V) → 𝐴 ∈ V)
65ancoms 459 . . . 4 ((ω ∈ V ∧ 𝐴 ⊆ ω) → 𝐴 ∈ V)
763adant3 1132 . . 3 ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐴 𝑥𝑦) → 𝐴 ∈ V)
8 eqid 2732 . . . . 5 (rec((𝑧 ∈ V ↦ (𝐴 ∖ suc 𝑧)), 𝐴) ↾ ω) = (rec((𝑧 ∈ V ↦ (𝐴 ∖ suc 𝑧)), 𝐴) ↾ ω)
98unblem4 9294 . . . 4 ((𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐴 𝑥𝑦) → (rec((𝑧 ∈ V ↦ (𝐴 ∖ suc 𝑧)), 𝐴) ↾ ω):ω–1-1𝐴)
1093adant1 1130 . . 3 ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐴 𝑥𝑦) → (rec((𝑧 ∈ V ↦ (𝐴 ∖ suc 𝑧)), 𝐴) ↾ ω):ω–1-1𝐴)
11 f1dom2g 8961 . . 3 ((ω ∈ V ∧ 𝐴 ∈ V ∧ (rec((𝑧 ∈ V ↦ (𝐴 ∖ suc 𝑧)), 𝐴) ↾ ω):ω–1-1𝐴) → ω ≼ 𝐴)
124, 7, 10, 11syl3anc 1371 . 2 ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐴 𝑥𝑦) → ω ≼ 𝐴)
13 sbth 9089 . 2 ((𝐴 ≼ ω ∧ ω ≼ 𝐴) → 𝐴 ≈ ω)
143, 12, 13syl2anc 584 1 ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐴 𝑥𝑦) → 𝐴 ≈ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087  wcel 2106  wral 3061  wrex 3070  Vcvv 3474  cdif 3944  wss 3947   cint 4949   class class class wbr 5147  cmpt 5230  cres 5677  suc csuc 6363  1-1wf1 6537  ωcom 7851  reccrdg 8405  cen 8932  cdom 8933
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-om 7852  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-en 8936  df-dom 8937
This theorem is referenced by:  unbnn2  9296  isfinite2  9297  unbnn3  9650
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