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Mirrors > Home > MPE Home > Th. List > unbnn | Structured version Visualization version GIF version |
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 9116 for a stronger version without the first assumption. (Contributed by NM, 3-Dec-2003.) |
Ref | Expression |
---|---|
unbnn | ⊢ ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → 𝐴 ≈ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssdomg 8549 | . . . 4 ⊢ (ω ∈ V → (𝐴 ⊆ ω → 𝐴 ≼ ω)) | |
2 | 1 | imp 409 | . . 3 ⊢ ((ω ∈ V ∧ 𝐴 ⊆ ω) → 𝐴 ≼ ω) |
3 | 2 | 3adant3 1128 | . 2 ⊢ ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → 𝐴 ≼ ω) |
4 | simp1 1132 | . . 3 ⊢ ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → ω ∈ V) | |
5 | ssexg 5220 | . . . . 5 ⊢ ((𝐴 ⊆ ω ∧ ω ∈ V) → 𝐴 ∈ V) | |
6 | 5 | ancoms 461 | . . . 4 ⊢ ((ω ∈ V ∧ 𝐴 ⊆ ω) → 𝐴 ∈ V) |
7 | 6 | 3adant3 1128 | . . 3 ⊢ ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → 𝐴 ∈ V) |
8 | eqid 2821 | . . . . 5 ⊢ (rec((𝑧 ∈ V ↦ ∩ (𝐴 ∖ suc 𝑧)), ∩ 𝐴) ↾ ω) = (rec((𝑧 ∈ V ↦ ∩ (𝐴 ∖ suc 𝑧)), ∩ 𝐴) ↾ ω) | |
9 | 8 | unblem4 8767 | . . . 4 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → (rec((𝑧 ∈ V ↦ ∩ (𝐴 ∖ suc 𝑧)), ∩ 𝐴) ↾ ω):ω–1-1→𝐴) |
10 | 9 | 3adant1 1126 | . . 3 ⊢ ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → (rec((𝑧 ∈ V ↦ ∩ (𝐴 ∖ suc 𝑧)), ∩ 𝐴) ↾ ω):ω–1-1→𝐴) |
11 | f1dom2g 8521 | . . 3 ⊢ ((ω ∈ V ∧ 𝐴 ∈ V ∧ (rec((𝑧 ∈ V ↦ ∩ (𝐴 ∖ suc 𝑧)), ∩ 𝐴) ↾ ω):ω–1-1→𝐴) → ω ≼ 𝐴) | |
12 | 4, 7, 10, 11 | syl3anc 1367 | . 2 ⊢ ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → ω ≼ 𝐴) |
13 | sbth 8631 | . 2 ⊢ ((𝐴 ≼ ω ∧ ω ≼ 𝐴) → 𝐴 ≈ ω) | |
14 | 3, 12, 13 | syl2anc 586 | 1 ⊢ ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → 𝐴 ≈ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 ∈ wcel 2110 ∀wral 3138 ∃wrex 3139 Vcvv 3495 ∖ cdif 3933 ⊆ wss 3936 ∩ cint 4869 class class class wbr 5059 ↦ cmpt 5139 ↾ cres 5552 suc csuc 6188 –1-1→wf1 6347 ωcom 7574 reccrdg 8039 ≈ cen 8500 ≼ cdom 8501 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-en 8504 df-dom 8505 |
This theorem is referenced by: unbnn2 8769 isfinite2 8770 unbnn3 9116 |
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