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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 2832 | . 2 ⊢ (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶) |
| 4 | 1, 3 | mtbir 323 | 1 ⊢ ¬ 𝐴 ∈ 𝐶 |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 |
| 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 |
| This theorem is referenced by: iprc 7933 tfr2b 8436 tz7.48-3 8484 pnfnre 11302 mnfnre 11304 prmrec 16960 00lsp 20979 goaln0 35398 bj-pinftynrr 37223 bj-minftynrr 37227 eliuniincex 45114 eliincex 45115 salgencntex 46358 nfermltl2rev 47730 |
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