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Mirrors > Home > MPE Home > Th. List > ring1ne0 | Structured version Visualization version GIF version |
Description: If a ring has at least two elements, its one and zero are different. (Contributed by AV, 13-Apr-2019.) |
Ref | Expression |
---|---|
ring1ne0.b | ⊢ 𝐵 = (Base‘𝑅) |
ring1ne0.u | ⊢ 1 = (1r‘𝑅) |
ring1ne0.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
ring1ne0 | ⊢ ((𝑅 ∈ Ring ∧ 1 < (♯‘𝐵)) → 1 ≠ 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ring1ne0.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
2 | 1 | fvexi 6851 | . . . 4 ⊢ 𝐵 ∈ V |
3 | hashgt12el 14249 | . . . 4 ⊢ ((𝐵 ∈ V ∧ 1 < (♯‘𝐵)) → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝑥 ≠ 𝑦) | |
4 | 2, 3 | mpan 688 | . . 3 ⊢ (1 < (♯‘𝐵) → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝑥 ≠ 𝑦) |
5 | 4 | adantl 482 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 1 < (♯‘𝐵)) → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝑥 ≠ 𝑦) |
6 | ring1ne0.u | . . . . . . . 8 ⊢ 1 = (1r‘𝑅) | |
7 | ring1ne0.z | . . . . . . . 8 ⊢ 0 = (0g‘𝑅) | |
8 | 1, 6, 7 | ring1eq0 19937 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ( 1 = 0 → 𝑥 = 𝑦)) |
9 | 8 | necon3d 2962 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ≠ 𝑦 → 1 ≠ 0 )) |
10 | 9 | 3expib 1122 | . . . . 5 ⊢ (𝑅 ∈ Ring → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ≠ 𝑦 → 1 ≠ 0 ))) |
11 | 10 | adantr 481 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 1 < (♯‘𝐵)) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ≠ 𝑦 → 1 ≠ 0 ))) |
12 | 11 | com3l 89 | . . 3 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ≠ 𝑦 → ((𝑅 ∈ Ring ∧ 1 < (♯‘𝐵)) → 1 ≠ 0 ))) |
13 | 12 | rexlimivv 3194 | . 2 ⊢ (∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝑥 ≠ 𝑦 → ((𝑅 ∈ Ring ∧ 1 < (♯‘𝐵)) → 1 ≠ 0 )) |
14 | 5, 13 | mpcom 38 | 1 ⊢ ((𝑅 ∈ Ring ∧ 1 < (♯‘𝐵)) → 1 ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2941 ∃wrex 3071 Vcvv 3443 class class class wbr 5103 ‘cfv 6491 1c1 10985 < clt 11122 ♯chash 14157 Basecbs 17017 0gc0g 17255 1rcur 19842 Ringcrg 19888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7662 ax-cnex 11040 ax-resscn 11041 ax-1cn 11042 ax-icn 11043 ax-addcl 11044 ax-addrcl 11045 ax-mulcl 11046 ax-mulrcl 11047 ax-mulcom 11048 ax-addass 11049 ax-mulass 11050 ax-distr 11051 ax-i2m1 11052 ax-1ne0 11053 ax-1rid 11054 ax-rnegex 11055 ax-rrecex 11056 ax-cnre 11057 ax-pre-lttri 11058 ax-pre-lttrn 11059 ax-pre-ltadd 11060 ax-pre-mulgt0 11061 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5528 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5585 df-we 5587 df-xp 5636 df-rel 5637 df-cnv 5638 df-co 5639 df-dm 5640 df-rn 5641 df-res 5642 df-ima 5643 df-pred 6249 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6443 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-riota 7305 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7793 df-1st 7911 df-2nd 7912 df-frecs 8179 df-wrecs 8210 df-recs 8284 df-rdg 8323 df-1o 8379 df-er 8581 df-en 8817 df-dom 8818 df-sdom 8819 df-fin 8820 df-card 9808 df-pnf 11124 df-mnf 11125 df-xr 11126 df-ltxr 11127 df-le 11128 df-sub 11320 df-neg 11321 df-nn 12087 df-2 12149 df-n0 12347 df-xnn0 12419 df-z 12433 df-uz 12696 df-fz 13353 df-hash 14158 df-sets 16970 df-slot 16988 df-ndx 17000 df-base 17018 df-plusg 17080 df-0g 17257 df-mgm 18431 df-sgrp 18480 df-mnd 18491 df-grp 18685 df-minusg 18686 df-mgp 19826 df-ur 19843 df-ring 19890 |
This theorem is referenced by: isnzr2hash 20657 01eq0ring 20665 el0ldep 46296 |
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