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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 3126 | . . 3 ⊢ (𝐴 = 𝐶 → (𝐵 ⊆ 𝐴 ↔ 𝐵 ⊆ 𝐶)) | |
4 | 3 | biimpa 294 | . 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ 𝐶) |
5 | 1, 2, 4 | sylancl 410 | 1 ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 ⊆ wss 3076 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-11 1485 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-in 3082 df-ss 3089 |
This theorem is referenced by: fssdm 5295 fndmdif 5533 fneqeql2 5537 fconst4m 5648 f1opw2 5984 ecss 6478 fopwdom 6738 ssenen 6753 phplem2 6755 fiintim 6825 casefun 6978 caseinj 6982 djufun 6997 djuinj 6999 nn0supp 9053 monoord2 10281 binom1dif 11288 cnpnei 12427 cnntri 12432 cnntr 12433 cncnp 12438 cndis 12449 txdis1cn 12486 hmeontr 12521 hmeoimaf1o 12522 dvcoapbr 12879 |
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