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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 9697 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 9039 | . . . 4 ⊢ (ω ∈ V → (𝐴 ⊆ ω → 𝐴 ≼ ω)) | |
2 | 1 | imp 406 | . . 3 ⊢ ((ω ∈ V ∧ 𝐴 ⊆ ω) → 𝐴 ≼ ω) |
3 | 2 | 3adant3 1131 | . 2 ⊢ ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → 𝐴 ≼ ω) |
4 | simp1 1135 | . . 3 ⊢ ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → ω ∈ V) | |
5 | ssexg 5329 | . . . . 5 ⊢ ((𝐴 ⊆ ω ∧ ω ∈ V) → 𝐴 ∈ V) | |
6 | 5 | ancoms 458 | . . . 4 ⊢ ((ω ∈ V ∧ 𝐴 ⊆ ω) → 𝐴 ∈ V) |
7 | 6 | 3adant3 1131 | . . 3 ⊢ ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → 𝐴 ∈ V) |
8 | eqid 2735 | . . . . 5 ⊢ (rec((𝑧 ∈ V ↦ ∩ (𝐴 ∖ suc 𝑧)), ∩ 𝐴) ↾ ω) = (rec((𝑧 ∈ V ↦ ∩ (𝐴 ∖ suc 𝑧)), ∩ 𝐴) ↾ ω) | |
9 | 8 | unblem4 9329 | . . . 4 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → (rec((𝑧 ∈ V ↦ ∩ (𝐴 ∖ suc 𝑧)), ∩ 𝐴) ↾ ω):ω–1-1→𝐴) |
10 | 9 | 3adant1 1129 | . . 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 9009 | . . 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 1370 | . 2 ⊢ ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → ω ≼ 𝐴) |
13 | sbth 9132 | . 2 ⊢ ((𝐴 ≼ ω ∧ ω ≼ 𝐴) → 𝐴 ≈ ω) | |
14 | 3, 12, 13 | syl2anc 584 | 1 ⊢ ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → 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 –1-1→wf1 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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