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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 6269 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴) | |
2 | ssel 3961 | . 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 2114 ⊆ wss 3936 suc csuc 6193 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-un 3941 df-in 3943 df-ss 3952 df-sn 4568 df-suc 6197 |
This theorem is referenced by: suc11 6294 ordelsuc 7535 ordsucelsuc 7537 oaordi 8172 nnaordi 8244 unbnn2 8775 ackbij1b 9661 ackbij2 9665 cflm 9672 isf32lem2 9776 indpi 10329 dfon2lem3 33030 |
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