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Mirrors > Home > ILE Home > Th. List > eqsstrdi | GIF version |
Description: A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
eqsstrdi.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
eqsstrdi.2 | ⊢ 𝐵 ⊆ 𝐶 |
Ref | Expression |
---|---|
eqsstrdi | ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqsstrdi.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | eqsstrdi.2 | . . 3 ⊢ 𝐵 ⊆ 𝐶 | |
3 | 2 | a1i 9 | . 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
4 | 1, 3 | eqsstrd 3133 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = 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: eqsstrrdi 3150 resasplitss 5302 fimacnv 5549 en2other2 7052 exmidfodomrlemim 7057 suplocexprlemex 7530 ennnfonelemkh 11925 toponsspwpwg 12189 ntrss2 12290 cnprcl2k 12375 reldvg 12817 bj-nntrans 13149 nninfsellemsuc 13208 |
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