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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 9651 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 8993 | . . . 4 ⊢ (ω ∈ V → (𝐴 ⊆ ω → 𝐴 ≼ ω)) | |
2 | 1 | imp 406 | . . 3 ⊢ ((ω ∈ V ∧ 𝐴 ⊆ ω) → 𝐴 ≼ ω) |
3 | 2 | 3adant3 1129 | . 2 ⊢ ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → 𝐴 ≼ ω) |
4 | simp1 1133 | . . 3 ⊢ ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → ω ∈ V) | |
5 | ssexg 5314 | . . . . 5 ⊢ ((𝐴 ⊆ ω ∧ ω ∈ V) → 𝐴 ∈ V) | |
6 | 5 | ancoms 458 | . . . 4 ⊢ ((ω ∈ V ∧ 𝐴 ⊆ ω) → 𝐴 ∈ V) |
7 | 6 | 3adant3 1129 | . . 3 ⊢ ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → 𝐴 ∈ V) |
8 | eqid 2724 | . . . . 5 ⊢ (rec((𝑧 ∈ V ↦ ∩ (𝐴 ∖ suc 𝑧)), ∩ 𝐴) ↾ ω) = (rec((𝑧 ∈ V ↦ ∩ (𝐴 ∖ suc 𝑧)), ∩ 𝐴) ↾ ω) | |
9 | 8 | unblem4 9295 | . . . 4 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → (rec((𝑧 ∈ V ↦ ∩ (𝐴 ∖ suc 𝑧)), ∩ 𝐴) ↾ ω):ω–1-1→𝐴) |
10 | 9 | 3adant1 1127 | . . 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 8962 | . . 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 1368 | . 2 ⊢ ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → ω ≼ 𝐴) |
13 | sbth 9090 | . 2 ⊢ ((𝐴 ≼ ω ∧ ω ≼ 𝐴) → 𝐴 ≈ ω) | |
14 | 3, 12, 13 | syl2anc 583 | 1 ⊢ ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → 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 –1-1→wf1 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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