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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 6852 | . . . 4 ⊢ 𝐵 ∈ V |
3 | hashgt12el 14250 | . . . 4 ⊢ ((𝐵 ∈ V ∧ 1 < (♯‘𝐵)) → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝑥 ≠ 𝑦) | |
4 | 2, 3 | mpan 689 | . . 3 ⊢ (1 < (♯‘𝐵) → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝑥 ≠ 𝑦) |
5 | 4 | adantl 483 | . 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 2963 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ≠ 𝑦 → 1 ≠ 0 )) |
10 | 9 | 3expib 1123 | . . . . 5 ⊢ (𝑅 ∈ Ring → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ≠ 𝑦 → 1 ≠ 0 ))) |
11 | 10 | adantr 482 | . . . 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 3195 | . 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 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2942 ∃wrex 3072 Vcvv 3444 class class class wbr 5104 ‘cfv 6492 1c1 10986 < clt 11123 ♯chash 14158 Basecbs 17018 0gc0g 17256 1rcur 19842 Ringcrg 19888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-om 7794 df-1st 7912 df-2nd 7913 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-1o 8380 df-er 8582 df-en 8818 df-dom 8819 df-sdom 8820 df-fin 8821 df-card 9809 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-nn 12088 df-2 12150 df-n0 12348 df-xnn0 12420 df-z 12434 df-uz 12697 df-fz 13354 df-hash 14159 df-sets 16971 df-slot 16989 df-ndx 17001 df-base 17019 df-plusg 17081 df-0g 17258 df-mgm 18432 df-sgrp 18481 df-mnd 18492 df-grp 18686 df-minusg 18687 df-mgp 19826 df-ur 19843 df-ring 19890 |
This theorem is referenced by: isnzr2hash 20657 01eq0ring 20665 el0ldep 46247 |
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