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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 8855 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 8289 | . . . 4 ⊢ (ω ∈ V → (𝐴 ⊆ ω → 𝐴 ≼ ω)) | |
2 | 1 | imp 397 | . . 3 ⊢ ((ω ∈ V ∧ 𝐴 ⊆ ω) → 𝐴 ≼ ω) |
3 | 2 | 3adant3 1123 | . 2 ⊢ ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → 𝐴 ≼ ω) |
4 | simp1 1127 | . . 3 ⊢ ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → ω ∈ V) | |
5 | ssexg 5043 | . . . . 5 ⊢ ((𝐴 ⊆ ω ∧ ω ∈ V) → 𝐴 ∈ V) | |
6 | 5 | ancoms 452 | . . . 4 ⊢ ((ω ∈ V ∧ 𝐴 ⊆ ω) → 𝐴 ∈ V) |
7 | 6 | 3adant3 1123 | . . 3 ⊢ ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → 𝐴 ∈ V) |
8 | eqid 2778 | . . . . 5 ⊢ (rec((𝑧 ∈ V ↦ ∩ (𝐴 ∖ suc 𝑧)), ∩ 𝐴) ↾ ω) = (rec((𝑧 ∈ V ↦ ∩ (𝐴 ∖ suc 𝑧)), ∩ 𝐴) ↾ ω) | |
9 | 8 | unblem4 8505 | . . . 4 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → (rec((𝑧 ∈ V ↦ ∩ (𝐴 ∖ suc 𝑧)), ∩ 𝐴) ↾ ω):ω–1-1→𝐴) |
10 | 9 | 3adant1 1121 | . . 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 8261 | . . 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 1439 | . 2 ⊢ ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → ω ≼ 𝐴) |
13 | sbth 8370 | . 2 ⊢ ((𝐴 ≼ ω ∧ ω ≼ 𝐴) → 𝐴 ≈ ω) | |
14 | 3, 12, 13 | syl2anc 579 | 1 ⊢ ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → 𝐴 ≈ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1071 ∈ wcel 2107 ∀wral 3090 ∃wrex 3091 Vcvv 3398 ∖ cdif 3789 ⊆ wss 3792 ∩ cint 4712 class class class wbr 4888 ↦ cmpt 4967 ↾ cres 5359 suc csuc 5980 –1-1→wf1 6134 ωcom 7345 reccrdg 7790 ≈ cen 8240 ≼ cdom 8241 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-om 7346 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-en 8244 df-dom 8245 |
This theorem is referenced by: unbnn2 8507 isfinite2 8508 unbnn3 8855 |
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