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Mirrors > Home > ILE Home > Th. List > eleqtri | GIF version |
Description: Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
eleqtr.1 | ⊢ 𝐴 ∈ 𝐵 |
eleqtr.2 | ⊢ 𝐵 = 𝐶 |
Ref | Expression |
---|---|
eleqtri | ⊢ 𝐴 ∈ 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleqtr.1 | . 2 ⊢ 𝐴 ∈ 𝐵 | |
2 | eleqtr.2 | . . 3 ⊢ 𝐵 = 𝐶 | |
3 | 2 | eleq2i 2207 | . 2 ⊢ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐶) |
4 | 1, 3 | mpbi 144 | 1 ⊢ 𝐴 ∈ 𝐶 |
Colors of variables: wff set class |
Syntax hints: = wceq 1332 ∈ wcel 1481 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1424 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-4 1488 ax-17 1507 ax-ial 1515 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-cleq 2133 df-clel 2136 |
This theorem is referenced by: eleqtrri 2216 3eltr3i 2221 prid2 3638 2eluzge0 9397 fz01or 9922 ef0lem 11403 ege2le3 11414 efgt1p2 11438 efgt1p 11439 phi1 11931 cnrehmeocntop 12801 dvcjbr 12880 |
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