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| Mirrors > Home > MPE Home > Th. List > nnenom | Structured version Visualization version GIF version | ||
| Description: The set of positive integers (as a subset of complex numbers) is equinumerous to omega (the set of finite ordinal numbers). (Contributed by NM, 31-Jul-2004.) (Revised by Mario Carneiro, 15-Sep-2013.) |
| Ref | Expression |
|---|---|
| nnenom | ⊢ ℕ ≈ ω |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | omex 9683 | . . 3 ⊢ ω ∈ V | |
| 2 | nn0ex 12532 | . . 3 ⊢ ℕ0 ∈ V | |
| 3 | eqid 2737 | . . . 4 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) | |
| 4 | 3 | hashgf1o 14012 | . . 3 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω):ω–1-1-onto→ℕ0 |
| 5 | f1oen2g 9009 | . . 3 ⊢ ((ω ∈ V ∧ ℕ0 ∈ V ∧ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω):ω–1-1-onto→ℕ0) → ω ≈ ℕ0) | |
| 6 | 1, 2, 4, 5 | mp3an 1463 | . 2 ⊢ ω ≈ ℕ0 |
| 7 | nn0ennn 14020 | . 2 ⊢ ℕ0 ≈ ℕ | |
| 8 | 6, 7 | entr2i 9049 | 1 ⊢ ℕ ≈ ω |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 Vcvv 3480 class class class wbr 5143 ↦ cmpt 5225 ↾ cres 5687 –1-1-onto→wf1o 6560 (class class class)co 7431 ωcom 7887 reccrdg 8449 ≈ cen 8982 0cc0 11155 1c1 11156 + caddc 11158 ℕcn 12266 ℕ0cn0 12526 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 |
| This theorem is referenced by: nnct 14022 supcvg 15892 xpnnen 16247 znnen 16248 qnnen 16249 rexpen 16264 aleph1re 16281 aleph1irr 16282 bitsf1 16483 unben 16947 odinf 19581 odhash 19592 cygctb 19910 1stcfb 23453 2ndcredom 23458 1stcelcls 23469 hauspwdom 23509 met1stc 24534 met2ndci 24535 re2ndc 24822 iscmet3 25327 ovolctb2 25527 ovolfi 25529 ovoliunlem3 25539 iunmbl2 25592 uniiccdif 25613 dyadmbl 25635 opnmblALT 25638 mbfimaopnlem 25690 itg2seq 25777 aannenlem3 26372 dirith2 27572 nmounbseqi 30796 nmobndseqi 30798 minvecolem5 30900 padct 32731 f1ocnt 32804 dmvlsiga 34130 sigapildsys 34163 volmeas 34232 omssubadd 34302 carsgclctunlem3 34322 poimirlem30 37657 poimirlem32 37659 mblfinlem1 37664 ovoliunnfl 37669 heiborlem3 37820 heibor 37828 lzenom 42781 fiphp3d 42830 irrapx1 42839 pellex 42846 nnfoctb 45053 zenom 45057 qenom 45372 ioonct 45550 subsaliuncl 46373 caragenunicl 46539 caratheodory 46543 ovnsubaddlem2 46586 |
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