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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 8902 | . . . . . 6 ⊢ ω ∈ On | |
2 | nnenom 13162 | . . . . . . 7 ⊢ ℕ ≈ ω | |
3 | 2 | ensymi 8355 | . . . . . 6 ⊢ ω ≈ ℕ |
4 | isnumi 9168 | . . . . . 6 ⊢ ((ω ∈ On ∧ ω ≈ ℕ) → ℕ ∈ dom card) | |
5 | 1, 3, 4 | mp2an 680 | . . . . 5 ⊢ ℕ ∈ dom card |
6 | xpnum 9173 | . . . . 5 ⊢ ((ℕ ∈ dom card ∧ ℕ ∈ dom card) → (ℕ × ℕ) ∈ dom card) | |
7 | 5, 5, 6 | mp2an 680 | . . . 4 ⊢ (ℕ × ℕ) ∈ dom card |
8 | subf 10687 | . . . . . . 7 ⊢ − :(ℂ × ℂ)⟶ℂ | |
9 | ffun 6345 | . . . . . . 7 ⊢ ( − :(ℂ × ℂ)⟶ℂ → Fun − ) | |
10 | 8, 9 | ax-mp 5 | . . . . . 6 ⊢ Fun − |
11 | nnsscn 11443 | . . . . . . . 8 ⊢ ℕ ⊆ ℂ | |
12 | xpss12 5419 | . . . . . . . 8 ⊢ ((ℕ ⊆ ℂ ∧ ℕ ⊆ ℂ) → (ℕ × ℕ) ⊆ (ℂ × ℂ)) | |
13 | 11, 11, 12 | mp2an 680 | . . . . . . 7 ⊢ (ℕ × ℕ) ⊆ (ℂ × ℂ) |
14 | 8 | fdmi 6352 | . . . . . . 7 ⊢ dom − = (ℂ × ℂ) |
15 | 13, 14 | sseqtr4i 3889 | . . . . . 6 ⊢ (ℕ × ℕ) ⊆ dom − |
16 | fores 6427 | . . . . . 6 ⊢ ((Fun − ∧ (ℕ × ℕ) ⊆ dom − ) → ( − ↾ (ℕ × ℕ)):(ℕ × ℕ)–onto→( − “ (ℕ × ℕ))) | |
17 | 10, 15, 16 | mp2an 680 | . . . . 5 ⊢ ( − ↾ (ℕ × ℕ)):(ℕ × ℕ)–onto→( − “ (ℕ × ℕ)) |
18 | dfz2 11811 | . . . . . 6 ⊢ ℤ = ( − “ (ℕ × ℕ)) | |
19 | foeq3 6415 | . . . . . 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 223 | . . . 4 ⊢ ( − ↾ (ℕ × ℕ)):(ℕ × ℕ)–onto→ℤ |
22 | fodomnum 9276 | . . . 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 15423 | . . 3 ⊢ (ℕ × ℕ) ≈ ℕ | |
25 | domentr 8364 | . . 3 ⊢ ((ℤ ≼ (ℕ × ℕ) ∧ (ℕ × ℕ) ≈ ℕ) → ℤ ≼ ℕ) | |
26 | 23, 24, 25 | mp2an 680 | . 2 ⊢ ℤ ≼ ℕ |
27 | zex 11801 | . . 3 ⊢ ℤ ∈ V | |
28 | nnssz 11813 | . . 3 ⊢ ℕ ⊆ ℤ | |
29 | ssdomg 8351 | . . 3 ⊢ (ℤ ∈ V → (ℕ ⊆ ℤ → ℕ ≼ ℤ)) | |
30 | 27, 28, 29 | mp2 9 | . 2 ⊢ ℕ ≼ ℤ |
31 | sbth 8432 | . 2 ⊢ ((ℤ ≼ ℕ ∧ ℕ ≼ ℤ) → ℤ ≈ ℕ) | |
32 | 26, 30, 31 | mp2an 680 | 1 ⊢ ℤ ≈ ℕ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1508 ∈ wcel 2051 Vcvv 3410 ⊆ wss 3824 class class class wbr 4926 × cxp 5402 dom cdm 5404 ↾ cres 5406 “ cima 5407 Oncon0 6027 Fun wfun 6180 ⟶wf 6182 –onto→wfo 6184 ωcom 7395 ≈ cen 8302 ≼ cdom 8303 cardccrd 9157 ℂcc 10332 − cmin 10669 ℕcn 11438 ℤcz 11792 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-rep 5046 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-inf2 8897 ax-cnex 10390 ax-resscn 10391 ax-1cn 10392 ax-icn 10393 ax-addcl 10394 ax-addrcl 10395 ax-mulcl 10396 ax-mulrcl 10397 ax-mulcom 10398 ax-addass 10399 ax-mulass 10400 ax-distr 10401 ax-i2m1 10402 ax-1ne0 10403 ax-1rid 10404 ax-rnegex 10405 ax-rrecex 10406 ax-cnre 10407 ax-pre-lttri 10408 ax-pre-lttrn 10409 ax-pre-ltadd 10410 ax-pre-mulgt0 10411 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-reu 3090 df-rmo 3091 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-pss 3840 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-uni 4710 df-int 4747 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-tr 5028 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-se 5364 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-isom 6195 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-om 7396 df-1st 7500 df-2nd 7501 df-wrecs 7749 df-recs 7811 df-rdg 7849 df-1o 7904 df-oadd 7908 df-omul 7909 df-er 8088 df-map 8207 df-en 8306 df-dom 8307 df-sdom 8308 df-fin 8309 df-oi 8768 df-card 9161 df-acn 9164 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-sub 10671 df-neg 10672 df-nn 11439 df-n0 11707 df-z 11793 df-uz 12058 |
This theorem is referenced by: qnnen 15425 odinf 18464 odhash 18473 cygctb 18779 iscmet3 23615 dyadmbl 23920 mbfsup 23984 dya2iocct 31216 zenom 40767 |
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