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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 2734 | . 2 ⊢ (𝐴 = 𝐵 → 𝐶 = 𝐶) | |
3 | 1, 2 | neleq12d 3052 | 1 ⊢ (𝐴 = 𝐵 → (𝐴 ∉ 𝐶 ↔ 𝐵 ∉ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∉ wnel 3047 |
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-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-cleq 2725 df-clel 2811 df-nel 3048 |
This theorem is referenced by: ruALT 9598 ssnn0fi 13950 cnpart 15187 sqrmo 15198 resqrtcl 15200 resqrtthlem 15201 sqrtneg 15214 sqreu 15307 sqrtthlem 15309 eqsqrtd 15314 ge2nprmge4 16638 prmgaplem7 16990 mgmnsgrpex 18812 sgrpnmndex 18813 iccpnfcnv 24460 griedg0prc 28552 nbgrssovtx 28649 rgrusgrprc 28877 rusgrprc 28878 rgrprcx 28880 frgrwopreglem4a 29594 xrge0iifcnv 32944 0nn0m1nnn0 34133 fpprel 46444 oddinmgm 46633 |
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