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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 4687 | . 2 ⊢ (Tr 𝐴 → ((𝐵 ∈ suc 𝐵 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐴)) | |
2 | sssucid 5719 | . . . . 5 ⊢ 𝐵 ⊆ suc 𝐵 | |
3 | ssexg 4732 | . . . . 5 ⊢ ((𝐵 ⊆ suc 𝐵 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ V) | |
4 | 2, 3 | mpan 702 | . . . 4 ⊢ (suc 𝐵 ∈ 𝐴 → 𝐵 ∈ V) |
5 | sucidg 5720 | . . . 4 ⊢ (𝐵 ∈ V → 𝐵 ∈ suc 𝐵) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (suc 𝐵 ∈ 𝐴 → 𝐵 ∈ suc 𝐵) |
7 | 6 | ancri 573 | . 2 ⊢ (suc 𝐵 ∈ 𝐴 → (𝐵 ∈ suc 𝐵 ∧ suc 𝐵 ∈ 𝐴)) |
8 | 1, 7 | impel 484 | 1 ⊢ ((Tr 𝐴 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 Tr wtr 4680 suc csuc 5642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-un 3545 df-in 3547 df-ss 3554 df-sn 4126 df-uni 4373 df-tr 4681 df-suc 5646 |
This theorem is referenced by: onuninsuci 6932 limsuc 6941 tz7.44-2 7390 cantnflt 8452 cantnfp1lem3 8460 cantnflem1b 8466 cantnflem1 8469 cnfcom 8480 axdc3lem2 9156 inar1 9476 bnj967 30269 limsuc2 36629 |
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