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Mirrors > Home > MPE Home > Th. List > dfz2 | Structured version Visualization version GIF version |
Description: Alternative definition of the integers, based on elz2 12073. (Contributed by Mario Carneiro, 16-May-2014.) |
Ref | Expression |
---|---|
dfz2 | ⊢ ℤ = ( − “ (ℕ × ℕ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elz2 12073 | . . 3 ⊢ (𝑥 ∈ ℤ ↔ ∃𝑦 ∈ ℕ ∃𝑧 ∈ ℕ 𝑥 = (𝑦 − 𝑧)) | |
2 | subf 10959 | . . . . 5 ⊢ − :(ℂ × ℂ)⟶ℂ | |
3 | ffn 6498 | . . . . 5 ⊢ ( − :(ℂ × ℂ)⟶ℂ → − Fn (ℂ × ℂ)) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ − Fn (ℂ × ℂ) |
5 | nnsscn 11714 | . . . . 5 ⊢ ℕ ⊆ ℂ | |
6 | xpss12 5534 | . . . . 5 ⊢ ((ℕ ⊆ ℂ ∧ ℕ ⊆ ℂ) → (ℕ × ℕ) ⊆ (ℂ × ℂ)) | |
7 | 5, 5, 6 | mp2an 692 | . . . 4 ⊢ (ℕ × ℕ) ⊆ (ℂ × ℂ) |
8 | ovelimab 7336 | . . . 4 ⊢ (( − Fn (ℂ × ℂ) ∧ (ℕ × ℕ) ⊆ (ℂ × ℂ)) → (𝑥 ∈ ( − “ (ℕ × ℕ)) ↔ ∃𝑦 ∈ ℕ ∃𝑧 ∈ ℕ 𝑥 = (𝑦 − 𝑧))) | |
9 | 4, 7, 8 | mp2an 692 | . . 3 ⊢ (𝑥 ∈ ( − “ (ℕ × ℕ)) ↔ ∃𝑦 ∈ ℕ ∃𝑧 ∈ ℕ 𝑥 = (𝑦 − 𝑧)) |
10 | 1, 9 | bitr4i 281 | . 2 ⊢ (𝑥 ∈ ℤ ↔ 𝑥 ∈ ( − “ (ℕ × ℕ))) |
11 | 10 | eqriv 2735 | 1 ⊢ ℤ = ( − “ (ℕ × ℕ)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1542 ∈ wcel 2113 ∃wrex 3054 ⊆ wss 3841 × cxp 5517 “ cima 5522 Fn wfn 6328 ⟶wf 6329 (class class class)co 7164 ℂcc 10606 − cmin 10941 ℕcn 11709 ℤcz 12055 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-1st 7707 df-2nd 7708 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-er 8313 df-en 8549 df-dom 8550 df-sdom 8551 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-nn 11710 df-n0 11970 df-z 12056 |
This theorem is referenced by: zexALT 12075 znnen 15650 |
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