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Mirrors > Home > MPE Home > Th. List > Mathboxes > gbepos | Structured version Visualization version Unicode version |
Description: Any even Goldbach number is positive. (Contributed by AV, 20-Jul-2020.) |
Ref | Expression |
---|---|
gbepos | GoldbachEven |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isgbe 41639 | . 2 GoldbachEven Even Odd Odd | |
2 | prmnn 15388 | . . . . . . . . 9 | |
3 | prmnn 15388 | . . . . . . . . 9 | |
4 | nnaddcl 11042 | . . . . . . . . 9 | |
5 | 2, 3, 4 | syl2an 494 | . . . . . . . 8 |
6 | eleq1 2689 | . . . . . . . 8 | |
7 | 5, 6 | syl5ibr 236 | . . . . . . 7 |
8 | 7 | 3ad2ant3 1084 | . . . . . 6 Odd Odd |
9 | 8 | com12 32 | . . . . 5 Odd Odd |
10 | 9 | a1i 11 | . . . 4 Even Odd Odd |
11 | 10 | rexlimdvv 3037 | . . 3 Even Odd Odd |
12 | 11 | imp 445 | . 2 Even Odd Odd |
13 | 1, 12 | sylbi 207 | 1 GoldbachEven |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wceq 1483 wcel 1990 wrex 2913 (class class class)co 6650 caddc 9939 cn 11020 cprime 15385 Even ceven 41537 Odd codd 41538 GoldbachEven cgbe 41633 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-addass 10001 ax-i2m1 10004 ax-1ne0 10005 ax-rrecex 10008 ax-cnre 10009 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-nn 11021 df-prm 15386 df-gbe 41636 |
This theorem is referenced by: gbege6 41653 |
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