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Mirrors > Home > MPE Home > Th. List > Mathboxes > 4even | Structured version Visualization version GIF version |
Description: 4 is an even number. (Contributed by AV, 23-Jul-2020.) |
Ref | Expression |
---|---|
4even | ⊢ 4 ∈ Even |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3odd 43369 | . 2 ⊢ 3 ∈ Odd | |
2 | df-4 11552 | . . 3 ⊢ 4 = (3 + 1) | |
3 | oddp1eveni 43302 | . . 3 ⊢ (3 ∈ Odd → (3 + 1) ∈ Even ) | |
4 | 2, 3 | syl5eqel 2886 | . 2 ⊢ (3 ∈ Odd → 4 ∈ Even ) |
5 | 1, 4 | ax-mp 5 | 1 ⊢ 4 ∈ Even |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2080 (class class class)co 7019 1c1 10387 + caddc 10389 3c3 11543 4c4 11544 Even ceven 43285 Odd codd 43286 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-sep 5097 ax-nul 5104 ax-pow 5160 ax-pr 5224 ax-un 7322 ax-resscn 10443 ax-1cn 10444 ax-icn 10445 ax-addcl 10446 ax-addrcl 10447 ax-mulcl 10448 ax-mulrcl 10449 ax-mulcom 10450 ax-addass 10451 ax-mulass 10452 ax-distr 10453 ax-i2m1 10454 ax-1ne0 10455 ax-1rid 10456 ax-rnegex 10457 ax-rrecex 10458 ax-cnre 10459 ax-pre-lttri 10460 ax-pre-lttrn 10461 ax-pre-ltadd 10462 ax-pre-mulgt0 10463 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ne 2984 df-nel 3090 df-ral 3109 df-rex 3110 df-reu 3111 df-rmo 3112 df-rab 3113 df-v 3438 df-sbc 3708 df-csb 3814 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-pss 3878 df-nul 4214 df-if 4384 df-pw 4457 df-sn 4475 df-pr 4477 df-tp 4479 df-op 4481 df-uni 4748 df-iun 4829 df-br 4965 df-opab 5027 df-mpt 5044 df-tr 5067 df-id 5351 df-eprel 5356 df-po 5365 df-so 5366 df-fr 5405 df-we 5407 df-xp 5452 df-rel 5453 df-cnv 5454 df-co 5455 df-dm 5456 df-rn 5457 df-res 5458 df-ima 5459 df-pred 6026 df-ord 6072 df-on 6073 df-lim 6074 df-suc 6075 df-iota 6192 df-fun 6230 df-fn 6231 df-f 6232 df-f1 6233 df-fo 6234 df-f1o 6235 df-fv 6236 df-riota 6980 df-ov 7022 df-oprab 7023 df-mpo 7024 df-om 7440 df-wrecs 7801 df-recs 7863 df-rdg 7901 df-er 8142 df-en 8361 df-dom 8362 df-sdom 8363 df-pnf 10526 df-mnf 10527 df-xr 10528 df-ltxr 10529 df-le 10530 df-sub 10721 df-neg 10722 df-div 11148 df-nn 11489 df-2 11550 df-3 11551 df-4 11552 df-n0 11748 df-z 11832 df-even 43287 df-odd 43288 |
This theorem is referenced by: 5odd 43371 |
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