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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 8423 | . . 3 ⊢ ω ∈ V | |
2 | nn0ex 11175 | . . 3 ⊢ ℕ0 ∈ V | |
3 | eqid 2610 | . . . 4 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) | |
4 | 3 | hashgf1o 12632 | . . 3 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω):ω–1-1-onto→ℕ0 |
5 | f1oen2g 7858 | . . 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 1416 | . 2 ⊢ ω ≈ ℕ0 |
7 | nn0ennn 12640 | . 2 ⊢ ℕ0 ≈ ℕ | |
8 | 6, 7 | entr2i 7897 | 1 ⊢ ℕ ≈ ω |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 Vcvv 3173 class class class wbr 4583 ↦ cmpt 4643 ↾ cres 5040 –1-1-onto→wf1o 5803 (class class class)co 6549 ωcom 6957 reccrdg 7392 ≈ cen 7838 0cc0 9815 1c1 9816 + caddc 9818 ℕcn 10897 ℕ0cn0 11169 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 |
This theorem is referenced by: nnct 12642 supcvg 14427 xpnnen 14778 znnen 14780 qnnen 14781 rexpen 14796 aleph1re 14813 aleph1irr 14814 bitsf1 15006 unben 15451 odinf 17803 odhash 17812 cygctb 18116 1stcfb 21058 2ndcredom 21063 1stcelcls 21074 hauspwdom 21114 met1stc 22136 met2ndci 22137 re2ndc 22412 iscmet3 22899 ovolctb2 23067 ovolfi 23069 ovoliunlem3 23079 iunmbl2 23132 uniiccdif 23152 dyadmbl 23174 opnmblALT 23177 mbfimaopnlem 23228 itg2seq 23315 aannenlem3 23889 dirith2 25017 nmounbseqi 27016 nmobndseqi 27018 minvecolem5 27121 padct 28885 f1ocnt 28946 dmvlsiga 29519 sigapildsys 29552 volmeas 29621 omssubadd 29689 carsgclctunlem3 29709 poimirlem30 32609 poimirlem32 32611 mblfinlem1 32616 ovoliunnfl 32621 heiborlem3 32782 heibor 32790 lzenom 36351 fiphp3d 36401 irrapx1 36410 pellex 36417 nnfoctb 38238 zenom 38244 qenom 38518 ioonct 38611 subsaliuncl 39252 caragenunicl 39414 caratheodory 39418 ovnsubaddlem2 39461 |
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