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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 3050 | 1 ⊢ (𝐴 = 𝐵 → (𝐴 ∉ 𝐶 ↔ 𝐵 ∉ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∉ wnel 3046 |
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 3047 |
This theorem is referenced by: ruALT 9544 ssnn0fi 13896 cnpart 15131 sqrmo 15142 resqrtcl 15144 resqrtthlem 15145 sqrtneg 15158 sqreu 15251 sqrtthlem 15253 eqsqrtd 15258 ge2nprmge4 16582 prmgaplem7 16934 mgmnsgrpex 18746 sgrpnmndex 18747 iccpnfcnv 24323 griedg0prc 28254 nbgrssovtx 28351 rgrusgrprc 28579 rusgrprc 28580 rgrprcx 28582 frgrwopreglem4a 29296 xrge0iifcnv 32571 0nn0m1nnn0 33760 fpprel 46006 oddinmgm 46195 |
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