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Mirrors > Home > ILE Home > Th. List > sseqtrrid | GIF version |
Description: Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
Ref | Expression |
---|---|
sseqtrrid.1 | ⊢ 𝐵 ⊆ 𝐴 |
sseqtrrid.2 | ⊢ (𝜑 → 𝐶 = 𝐴) |
Ref | Expression |
---|---|
sseqtrrid | ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseqtrrid.1 | . . 3 ⊢ 𝐵 ⊆ 𝐴 | |
2 | 1 | a1i 9 | . 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
3 | sseqtrrid.2 | . 2 ⊢ (𝜑 → 𝐶 = 𝐴) | |
4 | 2, 3 | sseqtrrd 3176 | 1 ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = 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: resdif 5448 fimacnv 5608 tfrlem5 6273 fsumsplit 11334 fprodsplitdc 11523 phimullem 12136 ennnfonelemss 12286 istopon 12558 sscls 12667 mopnfss 12994 |
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