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Mirrors > Home > ILE Home > Th. List > trsucss | GIF version |
Description: A member of the successor of a transitive class is a subclass of it. (Contributed by NM, 4-Oct-2003.) |
Ref | Expression |
---|---|
trsucss | ⊢ (Tr 𝐴 → (𝐵 ∈ suc 𝐴 → 𝐵 ⊆ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elsuci 4254 | . 2 ⊢ (𝐵 ∈ suc 𝐴 → (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)) | |
2 | trss 3967 | . . 3 ⊢ (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)) | |
3 | eqimss 3093 | . . . 4 ⊢ (𝐵 = 𝐴 → 𝐵 ⊆ 𝐴) | |
4 | 3 | a1i 9 | . . 3 ⊢ (Tr 𝐴 → (𝐵 = 𝐴 → 𝐵 ⊆ 𝐴)) |
5 | 2, 4 | jaod 675 | . 2 ⊢ (Tr 𝐴 → ((𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴) → 𝐵 ⊆ 𝐴)) |
6 | 1, 5 | syl5 32 | 1 ⊢ (Tr 𝐴 → (𝐵 ∈ suc 𝐴 → 𝐵 ⊆ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 667 = wceq 1296 ∈ wcel 1445 ⊆ wss 3013 Tr wtr 3958 suc csuc 4216 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-v 2635 df-un 3017 df-in 3019 df-ss 3026 df-sn 3472 df-uni 3676 df-tr 3959 df-suc 4222 |
This theorem is referenced by: onsucsssucr 4354 ordpwsucss 4411 nnsf 12600 nninfalllemn 12603 |
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