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Theorem unbnn 9330
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 9697 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 9039 . . . 4 (ω ∈ V → (𝐴 ⊆ ω → 𝐴 ≼ ω))
21imp 406 . . 3 ((ω ∈ V ∧ 𝐴 ⊆ ω) → 𝐴 ≼ ω)
323adant3 1131 . 2 ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐴 𝑥𝑦) → 𝐴 ≼ ω)
4 simp1 1135 . . 3 ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐴 𝑥𝑦) → ω ∈ V)
5 ssexg 5329 . . . . 5 ((𝐴 ⊆ ω ∧ ω ∈ V) → 𝐴 ∈ V)
65ancoms 458 . . . 4 ((ω ∈ V ∧ 𝐴 ⊆ ω) → 𝐴 ∈ V)
763adant3 1131 . . 3 ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐴 𝑥𝑦) → 𝐴 ∈ V)
8 eqid 2735 . . . . 5 (rec((𝑧 ∈ V ↦ (𝐴 ∖ suc 𝑧)), 𝐴) ↾ ω) = (rec((𝑧 ∈ V ↦ (𝐴 ∖ suc 𝑧)), 𝐴) ↾ ω)
98unblem4 9329 . . . 4 ((𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐴 𝑥𝑦) → (rec((𝑧 ∈ V ↦ (𝐴 ∖ suc 𝑧)), 𝐴) ↾ ω):ω–1-1𝐴)
1093adant1 1129 . . 3 ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐴 𝑥𝑦) → (rec((𝑧 ∈ V ↦ (𝐴 ∖ suc 𝑧)), 𝐴) ↾ ω):ω–1-1𝐴)
11 f1dom2g 9009 . . 3 ((ω ∈ V ∧ 𝐴 ∈ V ∧ (rec((𝑧 ∈ V ↦ (𝐴 ∖ suc 𝑧)), 𝐴) ↾ ω):ω–1-1𝐴) → ω ≼ 𝐴)
124, 7, 10, 11syl3anc 1370 . 2 ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐴 𝑥𝑦) → ω ≼ 𝐴)
13 sbth 9132 . 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 1086  wcel 2106  wral 3059  wrex 3068  Vcvv 3478  cdif 3960  wss 3963   cint 4951   class class class wbr 5148  cmpt 5231  cres 5691  suc csuc 6388  1-1wf1 6560  ωcom 7887  reccrdg 8448  cen 8981  cdom 8982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-en 8985  df-dom 8986
This theorem is referenced by:  unbnn2  9331  isfinite2  9332  unbnn3  9697
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