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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 3121 | . . 3 ⊢ (𝐴 = 𝐶 → (𝐵 ⊆ 𝐴 ↔ 𝐵 ⊆ 𝐶)) | |
4 | 3 | biimpa 294 | . 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ 𝐶) |
5 | 1, 2, 4 | sylancl 409 | 1 ⊢ (𝜑 → 𝐵 ⊆ 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: fssdm 5287 fndmdif 5525 fneqeql2 5529 fconst4m 5640 f1opw2 5976 ecss 6470 fopwdom 6730 ssenen 6745 phplem2 6747 fiintim 6817 casefun 6970 caseinj 6974 djufun 6989 djuinj 6991 nn0supp 9032 monoord2 10253 binom1dif 11259 cnpnei 12391 cnntri 12396 cnntr 12397 cncnp 12402 cndis 12413 txdis1cn 12450 hmeontr 12485 hmeoimaf1o 12486 dvcoapbr 12843 |
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