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Mirrors > Home > MPE Home > Th. List > eqneltri | Structured version Visualization version GIF version |
Description: If a class is not an element of another class, an equal class is also not an element. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
Ref | Expression |
---|---|
eqneltri.1 | ⊢ 𝐴 = 𝐵 |
eqneltri.2 | ⊢ ¬ 𝐵 ∈ 𝐶 |
Ref | Expression |
---|---|
eqneltri | ⊢ ¬ 𝐴 ∈ 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqneltri.2 | . 2 ⊢ ¬ 𝐵 ∈ 𝐶 | |
2 | eqneltri.1 | . . 3 ⊢ 𝐴 = 𝐵 | |
3 | 2 | eleq1i 2825 | . 2 ⊢ (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶) |
4 | 1, 3 | mtbir 323 | 1 ⊢ ¬ 𝐴 ∈ 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1542 ∈ wcel 2107 |
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 |
This theorem is referenced by: iprc 7904 tfr2b 8396 tz7.48-3 8444 pnfnre 11255 mnfnre 11257 prmrec 16855 00lsp 20592 goaln0 34384 bj-pinftynrr 36103 bj-minftynrr 36107 eliuniincex 43798 eliincex 43799 salgencntex 45059 nfermltl2rev 46411 |
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