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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 6008 | . 2 ⊢ (𝐵 ∈ suc 𝐴 → (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)) | |
2 | trss 4955 | . . 3 ⊢ (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)) | |
3 | eqimss 3854 | . . . 4 ⊢ (𝐵 = 𝐴 → 𝐵 ⊆ 𝐴) | |
4 | 3 | a1i 11 | . . 3 ⊢ (Tr 𝐴 → (𝐵 = 𝐴 → 𝐵 ⊆ 𝐴)) |
5 | 2, 4 | jaod 886 | . 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 874 = wceq 1653 ∈ wcel 2157 ⊆ wss 3770 Tr wtr 4946 suc csuc 5944 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-ext 2778 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ral 3095 df-v 3388 df-un 3775 df-in 3777 df-ss 3784 df-sn 4370 df-uni 4630 df-tr 4947 df-suc 5948 |
This theorem is referenced by: efgmnvl 18439 |
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