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| Mirrors > Home > MPE Home > Th. List > neleq1 | Structured version Visualization version GIF version | ||
| Description: Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.) (Proof shortened by Wolf Lammen, 25-Nov-2019.) |
| Ref | Expression |
|---|---|
| neleq1 | ⊢ (𝐴 = 𝐵 → (𝐴 ∉ 𝐶 ↔ 𝐵 ∉ 𝐶)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | id 22 | . 2 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵) | |
| 2 | eqidd 2738 | . 2 ⊢ (𝐴 = 𝐵 → 𝐶 = 𝐶) | |
| 3 | 1, 2 | neleq12d 3051 | 1 ⊢ (𝐴 = 𝐵 → (𝐴 ∉ 𝐶 ↔ 𝐵 ∉ 𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∉ wnel 3046 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-cleq 2729 df-clel 2816 df-nel 3047 |
| This theorem is referenced by: ruALT 9643 ssnn0fi 14026 cnpart 15279 sqrmo 15290 resqrtcl 15292 resqrtthlem 15293 sqrtneg 15306 sqreu 15399 sqrtthlem 15401 eqsqrtd 15406 ge2nprmge4 16738 prmgaplem7 17095 mgmnsgrpex 18944 sgrpnmndex 18945 iccpnfcnv 24975 griedg0prc 29281 nbgrssovtx 29378 rgrusgrprc 29607 rusgrprc 29608 rgrprcx 29610 frgrwopreglem4a 30329 xrge0iifcnv 33932 0nn0m1nnn0 35118 fpprel 47715 gpg5nbgrvtx03star 48036 gpg5nbgr3star 48037 oddinmgm 48091 |
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