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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 5267 | . 2 ⊢ (Tr 𝐴 → ((𝐵 ∈ suc 𝐵 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐴)) | |
2 | sssucid 6433 | . . . . 5 ⊢ 𝐵 ⊆ suc 𝐵 | |
3 | ssexg 5316 | . . . . 5 ⊢ ((𝐵 ⊆ suc 𝐵 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ V) | |
4 | 2, 3 | mpan 688 | . . . 4 ⊢ (suc 𝐵 ∈ 𝐴 → 𝐵 ∈ V) |
5 | sucidg 6434 | . . . 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 2106 Vcvv 3473 ⊆ wss 3944 Tr wtr 5258 suc csuc 6355 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 ax-sep 5292 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-rab 3432 df-v 3475 df-un 3949 df-in 3951 df-ss 3961 df-sn 4623 df-uni 4902 df-tr 5259 df-suc 6359 |
This theorem is referenced by: onuninsuci 7812 limsuc 7821 tz7.44-2 8389 cantnflt 9649 cantnfp1lem3 9657 cantnflem1b 9663 cantnflem1 9666 cnfcom 9677 axdc3lem2 10428 inar1 10752 bnj967 33787 limsuc2 41554 |
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