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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 9647 | . . . . . 6 ⊢ ω ∈ On | |
2 | nnenom 13952 | . . . . . . 7 ⊢ ℕ ≈ ω | |
3 | 2 | ensymi 9006 | . . . . . 6 ⊢ ω ≈ ℕ |
4 | isnumi 9947 | . . . . . 6 ⊢ ((ω ∈ On ∧ ω ≈ ℕ) → ℕ ∈ dom card) | |
5 | 1, 3, 4 | mp2an 689 | . . . . 5 ⊢ ℕ ∈ dom card |
6 | xpnum 9952 | . . . . 5 ⊢ ((ℕ ∈ dom card ∧ ℕ ∈ dom card) → (ℕ × ℕ) ∈ dom card) | |
7 | 5, 5, 6 | mp2an 689 | . . . 4 ⊢ (ℕ × ℕ) ∈ dom card |
8 | subf 11469 | . . . . . . 7 ⊢ − :(ℂ × ℂ)⟶ℂ | |
9 | ffun 6720 | . . . . . . 7 ⊢ ( − :(ℂ × ℂ)⟶ℂ → Fun − ) | |
10 | 8, 9 | ax-mp 5 | . . . . . 6 ⊢ Fun − |
11 | nnsscn 12224 | . . . . . . . 8 ⊢ ℕ ⊆ ℂ | |
12 | xpss12 5691 | . . . . . . . 8 ⊢ ((ℕ ⊆ ℂ ∧ ℕ ⊆ ℂ) → (ℕ × ℕ) ⊆ (ℂ × ℂ)) | |
13 | 11, 11, 12 | mp2an 689 | . . . . . . 7 ⊢ (ℕ × ℕ) ⊆ (ℂ × ℂ) |
14 | 8 | fdmi 6729 | . . . . . . 7 ⊢ dom − = (ℂ × ℂ) |
15 | 13, 14 | sseqtrri 4019 | . . . . . 6 ⊢ (ℕ × ℕ) ⊆ dom − |
16 | fores 6815 | . . . . . 6 ⊢ ((Fun − ∧ (ℕ × ℕ) ⊆ dom − ) → ( − ↾ (ℕ × ℕ)):(ℕ × ℕ)–onto→( − “ (ℕ × ℕ))) | |
17 | 10, 15, 16 | mp2an 689 | . . . . 5 ⊢ ( − ↾ (ℕ × ℕ)):(ℕ × ℕ)–onto→( − “ (ℕ × ℕ)) |
18 | dfz2 12584 | . . . . . 6 ⊢ ℤ = ( − “ (ℕ × ℕ)) | |
19 | foeq3 6803 | . . . . . 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 230 | . . . 4 ⊢ ( − ↾ (ℕ × ℕ)):(ℕ × ℕ)–onto→ℤ |
22 | fodomnum 10058 | . . . 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 16161 | . . 3 ⊢ (ℕ × ℕ) ≈ ℕ | |
25 | domentr 9015 | . . 3 ⊢ ((ℤ ≼ (ℕ × ℕ) ∧ (ℕ × ℕ) ≈ ℕ) → ℤ ≼ ℕ) | |
26 | 23, 24, 25 | mp2an 689 | . 2 ⊢ ℤ ≼ ℕ |
27 | zex 12574 | . . 3 ⊢ ℤ ∈ V | |
28 | nnssz 12587 | . . 3 ⊢ ℕ ⊆ ℤ | |
29 | ssdomg 9002 | . . 3 ⊢ (ℤ ∈ V → (ℕ ⊆ ℤ → ℕ ≼ ℤ)) | |
30 | 27, 28, 29 | mp2 9 | . 2 ⊢ ℕ ≼ ℤ |
31 | sbth 9099 | . 2 ⊢ ((ℤ ≼ ℕ ∧ ℕ ≼ ℤ) → ℤ ≈ ℕ) | |
32 | 26, 30, 31 | mp2an 689 | 1 ⊢ ℤ ≈ ℕ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1540 ∈ wcel 2105 Vcvv 3473 ⊆ wss 3948 class class class wbr 5148 × cxp 5674 dom cdm 5676 ↾ cres 5678 “ cima 5679 Oncon0 6364 Fun wfun 6537 ⟶wf 6539 –onto→wfo 6541 ωcom 7859 ≈ cen 8942 ≼ cdom 8943 cardccrd 9936 ℂcc 11114 − cmin 11451 ℕcn 12219 ℤcz 12565 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9642 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-oadd 8476 df-omul 8477 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-oi 9511 df-card 9940 df-acn 9943 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-n0 12480 df-z 12566 df-uz 12830 |
This theorem is referenced by: qnnen 16163 odinf 19476 odhash 19487 cygctb 19805 iscmet3 25054 dyadmbl 25362 mbfsup 25426 dya2iocct 33592 zenom 44053 |
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