| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 5268 | . 2 ⊢ (Tr 𝐴 → ((𝐵 ∈ suc 𝐵 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐴)) | |
| 2 | sssucid 6464 | . . . . 5 ⊢ 𝐵 ⊆ suc 𝐵 | |
| 3 | ssexg 5323 | . . . . 5 ⊢ ((𝐵 ⊆ suc 𝐵 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ V) | |
| 4 | 2, 3 | mpan 690 | . . . 4 ⊢ (suc 𝐵 ∈ 𝐴 → 𝐵 ∈ V) |
| 5 | sucidg 6465 | . . . 4 ⊢ (𝐵 ∈ V → 𝐵 ∈ suc 𝐵) | |
| 6 | 4, 5 | syl 17 | . . 3 ⊢ (suc 𝐵 ∈ 𝐴 → 𝐵 ∈ suc 𝐵) |
| 7 | 6 | ancri 549 | . 2 ⊢ (suc 𝐵 ∈ 𝐴 → (𝐵 ∈ suc 𝐵 ∧ suc 𝐵 ∈ 𝐴)) |
| 8 | 1, 7 | impel 505 | 1 ⊢ ((Tr 𝐴 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 Tr wtr 5259 suc csuc 6386 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-un 3956 df-in 3958 df-ss 3968 df-sn 4627 df-uni 4908 df-tr 5260 df-suc 6390 |
| This theorem is referenced by: onuninsuci 7861 limsuc 7870 tz7.44-2 8447 cantnflt 9712 cantnfp1lem3 9720 cantnflem1b 9726 cantnflem1 9729 cnfcom 9740 axdc3lem2 10491 inar1 10815 bnj967 34959 limsuc2 43053 |
| Copyright terms: Public domain | W3C validator |