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Mirrors > Home > ILE Home > Th. List > sseqtrid | GIF version |
Description: Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
Ref | Expression |
---|---|
sseqtrid.1 | ⊢ 𝐵 ⊆ 𝐴 |
sseqtrid.2 | ⊢ (𝜑 → 𝐴 = 𝐶) |
Ref | Expression |
---|---|
sseqtrid | ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseqtrid.2 | . 2 ⊢ (𝜑 → 𝐴 = 𝐶) | |
2 | sseqtrid.1 | . 2 ⊢ 𝐵 ⊆ 𝐴 | |
3 | sseq2 3171 | . . 3 ⊢ (𝐴 = 𝐶 → (𝐵 ⊆ 𝐴 ↔ 𝐵 ⊆ 𝐶)) | |
4 | 3 | biimpa 294 | . 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ 𝐶) |
5 | 1, 2, 4 | sylancl 411 | 1 ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 ⊆ wss 3121 |
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 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-11 1499 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-in 3127 df-ss 3134 |
This theorem is referenced by: fssdm 5362 fndmdif 5601 fneqeql2 5605 fconst4m 5716 f1opw2 6055 ecss 6554 fopwdom 6814 ssenen 6829 phplem2 6831 fiintim 6906 casefun 7062 caseinj 7066 djufun 7081 djuinj 7083 nn0supp 9187 monoord2 10433 binom1dif 11450 cnpnei 13013 cnntri 13018 cnntr 13019 cncnp 13024 cndis 13035 txdis1cn 13072 hmeontr 13107 hmeoimaf1o 13108 dvcoapbr 13465 |
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