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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 9684 | . . . . . 6 ⊢ ω ∈ On | |
2 | nnenom 14018 | . . . . . . 7 ⊢ ℕ ≈ ω | |
3 | 2 | ensymi 9043 | . . . . . 6 ⊢ ω ≈ ℕ |
4 | isnumi 9984 | . . . . . 6 ⊢ ((ω ∈ On ∧ ω ≈ ℕ) → ℕ ∈ dom card) | |
5 | 1, 3, 4 | mp2an 692 | . . . . 5 ⊢ ℕ ∈ dom card |
6 | xpnum 9989 | . . . . 5 ⊢ ((ℕ ∈ dom card ∧ ℕ ∈ dom card) → (ℕ × ℕ) ∈ dom card) | |
7 | 5, 5, 6 | mp2an 692 | . . . 4 ⊢ (ℕ × ℕ) ∈ dom card |
8 | subf 11508 | . . . . . . 7 ⊢ − :(ℂ × ℂ)⟶ℂ | |
9 | ffun 6740 | . . . . . . 7 ⊢ ( − :(ℂ × ℂ)⟶ℂ → Fun − ) | |
10 | 8, 9 | ax-mp 5 | . . . . . 6 ⊢ Fun − |
11 | nnsscn 12269 | . . . . . . . 8 ⊢ ℕ ⊆ ℂ | |
12 | xpss12 5704 | . . . . . . . 8 ⊢ ((ℕ ⊆ ℂ ∧ ℕ ⊆ ℂ) → (ℕ × ℕ) ⊆ (ℂ × ℂ)) | |
13 | 11, 11, 12 | mp2an 692 | . . . . . . 7 ⊢ (ℕ × ℕ) ⊆ (ℂ × ℂ) |
14 | 8 | fdmi 6748 | . . . . . . 7 ⊢ dom − = (ℂ × ℂ) |
15 | 13, 14 | sseqtrri 4033 | . . . . . 6 ⊢ (ℕ × ℕ) ⊆ dom − |
16 | fores 6831 | . . . . . 6 ⊢ ((Fun − ∧ (ℕ × ℕ) ⊆ dom − ) → ( − ↾ (ℕ × ℕ)):(ℕ × ℕ)–onto→( − “ (ℕ × ℕ))) | |
17 | 10, 15, 16 | mp2an 692 | . . . . 5 ⊢ ( − ↾ (ℕ × ℕ)):(ℕ × ℕ)–onto→( − “ (ℕ × ℕ)) |
18 | dfz2 12630 | . . . . . 6 ⊢ ℤ = ( − “ (ℕ × ℕ)) | |
19 | foeq3 6819 | . . . . . 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 231 | . . . 4 ⊢ ( − ↾ (ℕ × ℕ)):(ℕ × ℕ)–onto→ℤ |
22 | fodomnum 10095 | . . . 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 16244 | . . 3 ⊢ (ℕ × ℕ) ≈ ℕ | |
25 | domentr 9052 | . . 3 ⊢ ((ℤ ≼ (ℕ × ℕ) ∧ (ℕ × ℕ) ≈ ℕ) → ℤ ≼ ℕ) | |
26 | 23, 24, 25 | mp2an 692 | . 2 ⊢ ℤ ≼ ℕ |
27 | zex 12620 | . . 3 ⊢ ℤ ∈ V | |
28 | nnssz 12633 | . . 3 ⊢ ℕ ⊆ ℤ | |
29 | ssdomg 9039 | . . 3 ⊢ (ℤ ∈ V → (ℕ ⊆ ℤ → ℕ ≼ ℤ)) | |
30 | 27, 28, 29 | mp2 9 | . 2 ⊢ ℕ ≼ ℤ |
31 | sbth 9132 | . 2 ⊢ ((ℤ ≼ ℕ ∧ ℕ ≼ ℤ) → ℤ ≈ ℕ) | |
32 | 26, 30, 31 | mp2an 692 | 1 ⊢ ℤ ≈ ℕ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ⊆ wss 3963 class class class wbr 5148 × cxp 5687 dom cdm 5689 ↾ cres 5691 “ cima 5692 Oncon0 6386 Fun wfun 6557 ⟶wf 6559 –onto→wfo 6561 ωcom 7887 ≈ cen 8981 ≼ cdom 8982 cardccrd 9973 ℂcc 11151 − cmin 11490 ℕcn 12264 ℤcz 12611 |
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-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
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-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 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-se 5642 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-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-oadd 8509 df-omul 8510 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-oi 9548 df-card 9977 df-acn 9980 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 |
This theorem is referenced by: qnnen 16246 odinf 19596 odhash 19607 cygctb 19925 iscmet3 25341 dyadmbl 25649 mbfsup 25713 dya2iocct 34262 zenom 44992 |
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