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Mirrors > Home > MPE Home > Th. List > trsucss | Structured version Visualization version 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 6259 | . 2 ⊢ (𝐵 ∈ suc 𝐴 → (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)) | |
2 | trss 5183 | . . 3 ⊢ (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)) | |
3 | eqimss 4025 | . . . 4 ⊢ (𝐵 = 𝐴 → 𝐵 ⊆ 𝐴) | |
4 | 3 | a1i 11 | . . 3 ⊢ (Tr 𝐴 → (𝐵 = 𝐴 → 𝐵 ⊆ 𝐴)) |
5 | 2, 4 | jaod 855 | . 2 ⊢ (Tr 𝐴 → ((𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴) → 𝐵 ⊆ 𝐴)) |
6 | 1, 5 | syl5 34 | 1 ⊢ (Tr 𝐴 → (𝐵 ∈ suc 𝐴 → 𝐵 ⊆ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ⊆ wss 3938 Tr wtr 5174 suc csuc 6195 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-v 3498 df-un 3943 df-in 3945 df-ss 3954 df-sn 4570 df-uni 4841 df-tr 5175 df-suc 6199 |
This theorem is referenced by: efgmnvl 18842 |
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