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Mirrors > Home > MPE Home > Th. List > znnen | Structured version Visualization version GIF version |
Description: The set of integers and the set of positive integers are equinumerous. Exercise 1 of [Gleason] p. 140. (Contributed by NM, 31-Jul-2004.) (Proof shortened by Mario Carneiro, 13-Jun-2014.) |
Ref | Expression |
---|---|
znnen | ⊢ ℤ ≈ ℕ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omelon 8426 | . . . . . 6 ⊢ ω ∈ On | |
2 | nnenom 12641 | . . . . . . 7 ⊢ ℕ ≈ ω | |
3 | 2 | ensymi 7892 | . . . . . 6 ⊢ ω ≈ ℕ |
4 | isnumi 8655 | . . . . . 6 ⊢ ((ω ∈ On ∧ ω ≈ ℕ) → ℕ ∈ dom card) | |
5 | 1, 3, 4 | mp2an 704 | . . . . 5 ⊢ ℕ ∈ dom card |
6 | xpnum 8660 | . . . . 5 ⊢ ((ℕ ∈ dom card ∧ ℕ ∈ dom card) → (ℕ × ℕ) ∈ dom card) | |
7 | 5, 5, 6 | mp2an 704 | . . . 4 ⊢ (ℕ × ℕ) ∈ dom card |
8 | subf 10162 | . . . . . . 7 ⊢ − :(ℂ × ℂ)⟶ℂ | |
9 | ffun 5961 | . . . . . . 7 ⊢ ( − :(ℂ × ℂ)⟶ℂ → Fun − ) | |
10 | 8, 9 | ax-mp 5 | . . . . . 6 ⊢ Fun − |
11 | nnsscn 10902 | . . . . . . . 8 ⊢ ℕ ⊆ ℂ | |
12 | xpss12 5148 | . . . . . . . 8 ⊢ ((ℕ ⊆ ℂ ∧ ℕ ⊆ ℂ) → (ℕ × ℕ) ⊆ (ℂ × ℂ)) | |
13 | 11, 11, 12 | mp2an 704 | . . . . . . 7 ⊢ (ℕ × ℕ) ⊆ (ℂ × ℂ) |
14 | 8 | fdmi 5965 | . . . . . . 7 ⊢ dom − = (ℂ × ℂ) |
15 | 13, 14 | sseqtr4i 3601 | . . . . . 6 ⊢ (ℕ × ℕ) ⊆ dom − |
16 | fores 6037 | . . . . . 6 ⊢ ((Fun − ∧ (ℕ × ℕ) ⊆ dom − ) → ( − ↾ (ℕ × ℕ)):(ℕ × ℕ)–onto→( − “ (ℕ × ℕ))) | |
17 | 10, 15, 16 | mp2an 704 | . . . . 5 ⊢ ( − ↾ (ℕ × ℕ)):(ℕ × ℕ)–onto→( − “ (ℕ × ℕ)) |
18 | dfz2 11272 | . . . . . 6 ⊢ ℤ = ( − “ (ℕ × ℕ)) | |
19 | foeq3 6026 | . . . . . 6 ⊢ (ℤ = ( − “ (ℕ × ℕ)) → (( − ↾ (ℕ × ℕ)):(ℕ × ℕ)–onto→ℤ ↔ ( − ↾ (ℕ × ℕ)):(ℕ × ℕ)–onto→( − “ (ℕ × ℕ)))) | |
20 | 18, 19 | ax-mp 5 | . . . . 5 ⊢ (( − ↾ (ℕ × ℕ)):(ℕ × ℕ)–onto→ℤ ↔ ( − ↾ (ℕ × ℕ)):(ℕ × ℕ)–onto→( − “ (ℕ × ℕ))) |
21 | 17, 20 | mpbir 220 | . . . 4 ⊢ ( − ↾ (ℕ × ℕ)):(ℕ × ℕ)–onto→ℤ |
22 | fodomnum 8763 | . . . 4 ⊢ ((ℕ × ℕ) ∈ dom card → (( − ↾ (ℕ × ℕ)):(ℕ × ℕ)–onto→ℤ → ℤ ≼ (ℕ × ℕ))) | |
23 | 7, 21, 22 | mp2 9 | . . 3 ⊢ ℤ ≼ (ℕ × ℕ) |
24 | xpnnen 14778 | . . 3 ⊢ (ℕ × ℕ) ≈ ℕ | |
25 | domentr 7901 | . . 3 ⊢ ((ℤ ≼ (ℕ × ℕ) ∧ (ℕ × ℕ) ≈ ℕ) → ℤ ≼ ℕ) | |
26 | 23, 24, 25 | mp2an 704 | . 2 ⊢ ℤ ≼ ℕ |
27 | zex 11263 | . . 3 ⊢ ℤ ∈ V | |
28 | nnssz 11274 | . . 3 ⊢ ℕ ⊆ ℤ | |
29 | ssdomg 7887 | . . 3 ⊢ (ℤ ∈ V → (ℕ ⊆ ℤ → ℕ ≼ ℤ)) | |
30 | 27, 28, 29 | mp2 9 | . 2 ⊢ ℕ ≼ ℤ |
31 | sbth 7965 | . 2 ⊢ ((ℤ ≼ ℕ ∧ ℕ ≼ ℤ) → ℤ ≈ ℕ) | |
32 | 26, 30, 31 | mp2an 704 | 1 ⊢ ℤ ≈ ℕ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 class class class wbr 4583 × cxp 5036 dom cdm 5038 ↾ cres 5040 “ cima 5041 Oncon0 5640 Fun wfun 5798 ⟶wf 5800 –onto→wfo 5802 ωcom 6957 ≈ cen 7838 ≼ cdom 7839 cardccrd 8644 ℂcc 9813 − cmin 10145 ℕcn 10897 ℤcz 11254 |
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-rep 4699 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-rmo 2904 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-int 4411 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-se 4998 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-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-omul 7452 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-oi 8298 df-card 8648 df-acn 8651 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: qnnen 14781 odinf 17803 odhash 17812 cygctb 18116 iscmet3 22899 dyadmbl 23174 mbfsup 23237 dya2iocct 29669 zenom 38244 |
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