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Mirrors > Home > MPE Home > Th. List > sucssel | Structured version Visualization version GIF version |
Description: A set whose successor is a subset of another class is a member of that class. (Contributed by NM, 16-Sep-1995.) |
Ref | Expression |
---|---|
sucssel | ⊢ (𝐴 ∈ 𝑉 → (suc 𝐴 ⊆ 𝐵 → 𝐴 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sucidg 6237 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴) | |
2 | ssel 3908 | . 2 ⊢ (suc 𝐴 ⊆ 𝐵 → (𝐴 ∈ suc 𝐴 → 𝐴 ∈ 𝐵)) | |
3 | 1, 2 | syl5com 31 | 1 ⊢ (𝐴 ∈ 𝑉 → (suc 𝐴 ⊆ 𝐵 → 𝐴 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 ⊆ wss 3881 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 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-suc 6165 |
This theorem is referenced by: suc11 6262 ordelsuc 7515 ordsucelsuc 7517 oaordi 8155 nnaordi 8227 unbnn2 8759 ackbij1b 9650 ackbij2 9654 cflm 9661 isf32lem2 9765 indpi 10318 dfon2lem3 33143 |
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