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Mirrors > Home > ILE Home > Th. List > sseqtri | GIF version |
Description: Substitution of equality into a subclass relationship. (Contributed by NM, 28-Jul-1995.) |
Ref | Expression |
---|---|
sseqtr.1 | ⊢ 𝐴 ⊆ 𝐵 |
sseqtr.2 | ⊢ 𝐵 = 𝐶 |
Ref | Expression |
---|---|
sseqtri | ⊢ 𝐴 ⊆ 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseqtr.1 | . 2 ⊢ 𝐴 ⊆ 𝐵 | |
2 | sseqtr.2 | . . 3 ⊢ 𝐵 = 𝐶 | |
3 | 2 | sseq2i 3124 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐶) |
4 | 1, 3 | mpbi 144 | 1 ⊢ 𝐴 ⊆ 𝐶 |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ⊆ wss 3071 |
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 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-11 1484 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-in 3077 df-ss 3084 |
This theorem is referenced by: sseqtrri 3132 eqimssi 3153 abssi 3172 ssun2 3240 inssddif 3317 difdifdirss 3447 pwundifss 4207 unixpss 4652 0ima 4899 sbthlem7 6851 toponsspwpwg 12189 eltg4i 12224 ntrss2 12290 isopn3 12294 tgioo 12715 dvfvalap 12819 dvcnp2cntop 12832 |
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