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Mirrors > Home > MPE Home > Th. List > trsuc | Structured version Visualization version GIF version |
Description: A set whose successor belongs to a transitive class also belongs. (Contributed by NM, 5-Sep-2003.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
Ref | Expression |
---|---|
trsuc | ⊢ ((Tr 𝐴 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trel 5143 | . 2 ⊢ (Tr 𝐴 → ((𝐵 ∈ suc 𝐵 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐴)) | |
2 | sssucid 6236 | . . . . 5 ⊢ 𝐵 ⊆ suc 𝐵 | |
3 | ssexg 5191 | . . . . 5 ⊢ ((𝐵 ⊆ suc 𝐵 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ V) | |
4 | 2, 3 | mpan 689 | . . . 4 ⊢ (suc 𝐵 ∈ 𝐴 → 𝐵 ∈ V) |
5 | sucidg 6237 | . . . 4 ⊢ (𝐵 ∈ V → 𝐵 ∈ suc 𝐵) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (suc 𝐵 ∈ 𝐴 → 𝐵 ∈ suc 𝐵) |
7 | 6 | ancri 553 | . 2 ⊢ (suc 𝐵 ∈ 𝐴 → (𝐵 ∈ suc 𝐵 ∧ suc 𝐵 ∈ 𝐴)) |
8 | 1, 7 | impel 509 | 1 ⊢ ((Tr 𝐴 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 Vcvv 3441 ⊆ wss 3881 Tr wtr 5136 suc csuc 6161 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 ax-sep 5167 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-uni 4801 df-tr 5137 df-suc 6165 |
This theorem is referenced by: onuninsuci 7535 limsuc 7544 tz7.44-2 8026 cantnflt 9119 cantnfp1lem3 9127 cantnflem1b 9133 cantnflem1 9136 cnfcom 9147 axdc3lem2 9862 inar1 10186 bnj967 32327 limsuc2 39985 |
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