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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 5360 fndmdif 5598 fneqeql2 5602 fconst4m 5713 f1opw2 6052 ecss 6550 fopwdom 6810 ssenen 6825 phplem2 6827 fiintim 6902 casefun 7058 caseinj 7062 djufun 7077 djuinj 7079 nn0supp 9174 monoord2 10420 binom1dif 11437 cnpnei 12934 cnntri 12939 cnntr 12940 cncnp 12945 cndis 12956 txdis1cn 12993 hmeontr 13028 hmeoimaf1o 13029 dvcoapbr 13386 |
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