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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 5170 | . 2 ⊢ (Tr 𝐴 → ((𝐵 ∈ suc 𝐵 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐴)) | |
2 | sssucid 6261 | . . . . 5 ⊢ 𝐵 ⊆ suc 𝐵 | |
3 | ssexg 5218 | . . . . 5 ⊢ ((𝐵 ⊆ suc 𝐵 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ V) | |
4 | 2, 3 | mpan 686 | . . . 4 ⊢ (suc 𝐵 ∈ 𝐴 → 𝐵 ∈ V) |
5 | sucidg 6262 | . . . 4 ⊢ (𝐵 ∈ V → 𝐵 ∈ suc 𝐵) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (suc 𝐵 ∈ 𝐴 → 𝐵 ∈ suc 𝐵) |
7 | 6 | ancri 550 | . 2 ⊢ (suc 𝐵 ∈ 𝐴 → (𝐵 ∈ suc 𝐵 ∧ suc 𝐵 ∈ 𝐴)) |
8 | 1, 7 | impel 506 | 1 ⊢ ((Tr 𝐴 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2105 Vcvv 3492 ⊆ wss 3933 Tr wtr 5163 suc csuc 6186 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-un 3938 df-in 3940 df-ss 3949 df-sn 4558 df-uni 4831 df-tr 5164 df-suc 6190 |
This theorem is referenced by: onuninsuci 7544 limsuc 7553 tz7.44-2 8032 cantnflt 9123 cantnfp1lem3 9131 cantnflem1b 9137 cantnflem1 9140 cnfcom 9151 axdc3lem2 9861 inar1 10185 bnj967 32116 limsuc2 39519 |
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