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Mirrors > Home > MPE Home > Th. List > subne0d | Structured version Visualization version GIF version |
Description: Two unequal numbers have nonzero difference. (Contributed by Mario Carneiro, 1-Jan-2017.) |
Ref | Expression |
---|---|
negidd.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
pncand.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
subne0d.3 | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
Ref | Expression |
---|---|
subne0d | ⊢ (𝜑 → (𝐴 − 𝐵) ≠ 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subne0d.3 | . 2 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
2 | negidd.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
3 | pncand.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
4 | subeq0 10186 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) = 0 ↔ 𝐴 = 𝐵)) | |
5 | 2, 3, 4 | syl2anc 691 | . . 3 ⊢ (𝜑 → ((𝐴 − 𝐵) = 0 ↔ 𝐴 = 𝐵)) |
6 | 5 | necon3bid 2826 | . 2 ⊢ (𝜑 → ((𝐴 − 𝐵) ≠ 0 ↔ 𝐴 ≠ 𝐵)) |
7 | 1, 6 | mpbird 246 | 1 ⊢ (𝜑 → (𝐴 − 𝐵) ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 (class class class)co 6549 ℂcc 9813 0cc0 9815 − cmin 10145 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-ltxr 9958 df-sub 10147 |
This theorem is referenced by: modsumfzodifsn 12605 abssubne0 13904 rlimuni 14129 climuni 14131 pwm1geoser 14439 evth 22566 dvlem 23466 dvconst 23486 dvid 23487 dvcnp2 23489 dvaddbr 23507 dvmulbr 23508 dvcobr 23515 dvcjbr 23518 dvrec 23524 dvcnvlem 23543 dvferm2lem 23553 taylthlem2 23932 ulmdvlem1 23958 ang180lem4 24342 ang180lem5 24343 ang180 24344 isosctrlem3 24350 isosctr 24351 ssscongptld 24352 angpieqvdlem 24355 angpieqvdlem2 24356 angpined 24357 angpieqvd 24358 chordthmlem 24359 chordthmlem2 24360 heron 24365 asinlem 24395 lgamgulmlem2 24556 lgamgulmlem3 24557 ttgcontlem1 25565 brbtwn2 25585 axcontlem8 25651 2sqmod 28979 signsvtn0 29973 unbdqndv2lem2 31671 bj-bary1lem 32337 bj-bary1lem1 32338 bj-bary1 32339 pellexlem6 36416 jm2.26lem3 36586 areaquad 36821 bcc0 37561 bccm1k 37563 abssubrp 38428 lptre2pt 38707 limclner 38718 fperdvper 38808 stoweidlem23 38916 wallispilem4 38961 wallispi 38963 wallispi2lem1 38964 wallispi2lem2 38965 wallispi2 38966 stirlinglem5 38971 fourierdlem4 39004 fourierdlem42 39042 fourierdlem74 39073 fourierdlem75 39074 fouriersw 39124 sigardiv 39699 sigarcol 39702 sharhght 39703 |
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