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Mirrors > Home > MPE Home > Th. List > Mathboxes > 1neven | Structured version Visualization version GIF version |
Description: 1 is not an even integer. (Contributed by AV, 12-Feb-2020.) |
Ref | Expression |
---|---|
2zrng.e | ⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} |
Ref | Expression |
---|---|
1neven | ⊢ 1 ∉ 𝐸 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | halfnz 12061 | . . . . . . 7 ⊢ ¬ (1 / 2) ∈ ℤ | |
2 | eleq1a 2908 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ → ((1 / 2) = 𝑥 → (1 / 2) ∈ ℤ)) | |
3 | 1, 2 | mtoi 201 | . . . . . 6 ⊢ (𝑥 ∈ ℤ → ¬ (1 / 2) = 𝑥) |
4 | 1cnd 10636 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ → 1 ∈ ℂ) | |
5 | zcn 11987 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
6 | 2cnne0 11848 | . . . . . . . 8 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
7 | 6 | a1i 11 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ → (2 ∈ ℂ ∧ 2 ≠ 0)) |
8 | divmul2 11302 | . . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → ((1 / 2) = 𝑥 ↔ 1 = (2 · 𝑥))) | |
9 | 4, 5, 7, 8 | syl3anc 1367 | . . . . . 6 ⊢ (𝑥 ∈ ℤ → ((1 / 2) = 𝑥 ↔ 1 = (2 · 𝑥))) |
10 | 3, 9 | mtbid 326 | . . . . 5 ⊢ (𝑥 ∈ ℤ → ¬ 1 = (2 · 𝑥)) |
11 | 10 | nrex 3269 | . . . 4 ⊢ ¬ ∃𝑥 ∈ ℤ 1 = (2 · 𝑥) |
12 | 11 | intnan 489 | . . 3 ⊢ ¬ (1 ∈ ℤ ∧ ∃𝑥 ∈ ℤ 1 = (2 · 𝑥)) |
13 | eqeq1 2825 | . . . . 5 ⊢ (𝑧 = 1 → (𝑧 = (2 · 𝑥) ↔ 1 = (2 · 𝑥))) | |
14 | 13 | rexbidv 3297 | . . . 4 ⊢ (𝑧 = 1 → (∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥) ↔ ∃𝑥 ∈ ℤ 1 = (2 · 𝑥))) |
15 | 2zrng.e | . . . 4 ⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} | |
16 | 14, 15 | elrab2 3683 | . . 3 ⊢ (1 ∈ 𝐸 ↔ (1 ∈ ℤ ∧ ∃𝑥 ∈ ℤ 1 = (2 · 𝑥))) |
17 | 12, 16 | mtbir 325 | . 2 ⊢ ¬ 1 ∈ 𝐸 |
18 | 17 | nelir 3126 | 1 ⊢ 1 ∉ 𝐸 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∉ wnel 3123 ∃wrex 3139 {crab 3142 (class class class)co 7156 ℂcc 10535 0cc0 10537 1c1 10538 · cmul 10542 / cdiv 11297 2c2 11693 ℤcz 11982 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-n0 11899 df-z 11983 |
This theorem is referenced by: 2zrngnmlid 44240 2zrngnmrid 44241 |
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