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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 3164 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐶) |
4 | 1, 3 | mpbi 144 | 1 ⊢ 𝐴 ⊆ 𝐶 |
Colors of variables: wff set class |
Syntax hints: = wceq 1342 ⊆ wss 3111 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-11 1493 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-in 3117 df-ss 3124 |
This theorem is referenced by: sseqtrri 3172 eqimssi 3193 abssi 3212 ssun2 3281 inssddif 3358 difdifdirss 3488 ifidss 3530 pwundifss 4257 unixpss 4711 0ima 4958 sbthlem7 6919 ssnnctlemct 12322 toponsspwpwg 12567 eltg4i 12602 ntrss2 12668 isopn3 12672 tgioo 13093 dvfvalap 13197 dvcnp2cntop 13210 |
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