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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 9111 | . . . . . 6 ⊢ ω ∈ On | |
2 | nnenom 13351 | . . . . . . 7 ⊢ ℕ ≈ ω | |
3 | 2 | ensymi 8561 | . . . . . 6 ⊢ ω ≈ ℕ |
4 | isnumi 9377 | . . . . . 6 ⊢ ((ω ∈ On ∧ ω ≈ ℕ) → ℕ ∈ dom card) | |
5 | 1, 3, 4 | mp2an 690 | . . . . 5 ⊢ ℕ ∈ dom card |
6 | xpnum 9382 | . . . . 5 ⊢ ((ℕ ∈ dom card ∧ ℕ ∈ dom card) → (ℕ × ℕ) ∈ dom card) | |
7 | 5, 5, 6 | mp2an 690 | . . . 4 ⊢ (ℕ × ℕ) ∈ dom card |
8 | subf 10890 | . . . . . . 7 ⊢ − :(ℂ × ℂ)⟶ℂ | |
9 | ffun 6519 | . . . . . . 7 ⊢ ( − :(ℂ × ℂ)⟶ℂ → Fun − ) | |
10 | 8, 9 | ax-mp 5 | . . . . . 6 ⊢ Fun − |
11 | nnsscn 11645 | . . . . . . . 8 ⊢ ℕ ⊆ ℂ | |
12 | xpss12 5572 | . . . . . . . 8 ⊢ ((ℕ ⊆ ℂ ∧ ℕ ⊆ ℂ) → (ℕ × ℕ) ⊆ (ℂ × ℂ)) | |
13 | 11, 11, 12 | mp2an 690 | . . . . . . 7 ⊢ (ℕ × ℕ) ⊆ (ℂ × ℂ) |
14 | 8 | fdmi 6526 | . . . . . . 7 ⊢ dom − = (ℂ × ℂ) |
15 | 13, 14 | sseqtrri 4006 | . . . . . 6 ⊢ (ℕ × ℕ) ⊆ dom − |
16 | fores 6602 | . . . . . 6 ⊢ ((Fun − ∧ (ℕ × ℕ) ⊆ dom − ) → ( − ↾ (ℕ × ℕ)):(ℕ × ℕ)–onto→( − “ (ℕ × ℕ))) | |
17 | 10, 15, 16 | mp2an 690 | . . . . 5 ⊢ ( − ↾ (ℕ × ℕ)):(ℕ × ℕ)–onto→( − “ (ℕ × ℕ)) |
18 | dfz2 12003 | . . . . . 6 ⊢ ℤ = ( − “ (ℕ × ℕ)) | |
19 | foeq3 6590 | . . . . . 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 233 | . . . 4 ⊢ ( − ↾ (ℕ × ℕ)):(ℕ × ℕ)–onto→ℤ |
22 | fodomnum 9485 | . . . 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 15566 | . . 3 ⊢ (ℕ × ℕ) ≈ ℕ | |
25 | domentr 8570 | . . 3 ⊢ ((ℤ ≼ (ℕ × ℕ) ∧ (ℕ × ℕ) ≈ ℕ) → ℤ ≼ ℕ) | |
26 | 23, 24, 25 | mp2an 690 | . 2 ⊢ ℤ ≼ ℕ |
27 | zex 11993 | . . 3 ⊢ ℤ ∈ V | |
28 | nnssz 12005 | . . 3 ⊢ ℕ ⊆ ℤ | |
29 | ssdomg 8557 | . . 3 ⊢ (ℤ ∈ V → (ℕ ⊆ ℤ → ℕ ≼ ℤ)) | |
30 | 27, 28, 29 | mp2 9 | . 2 ⊢ ℕ ≼ ℤ |
31 | sbth 8639 | . 2 ⊢ ((ℤ ≼ ℕ ∧ ℕ ≼ ℤ) → ℤ ≈ ℕ) | |
32 | 26, 30, 31 | mp2an 690 | 1 ⊢ ℤ ≈ ℕ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ⊆ wss 3938 class class class wbr 5068 × cxp 5555 dom cdm 5557 ↾ cres 5559 “ cima 5560 Oncon0 6193 Fun wfun 6351 ⟶wf 6353 –onto→wfo 6355 ωcom 7582 ≈ cen 8508 ≼ cdom 8509 cardccrd 9366 ℂcc 10537 − cmin 10872 ℕcn 11640 ℤcz 11984 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-omul 8109 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-oi 8976 df-card 9370 df-acn 9373 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 |
This theorem is referenced by: qnnen 15568 odinf 18692 odhash 18701 cygctb 19014 iscmet3 23898 dyadmbl 24203 mbfsup 24267 dya2iocct 31540 zenom 41321 |
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