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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 6465 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴) | |
| 2 | ssel 3977 | . 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 2108 ⊆ wss 3951 suc csuc 6386 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-ss 3968 df-sn 4627 df-suc 6390 |
| This theorem is referenced by: suc11 6491 ordelsuc 7840 ordsucelsuc 7842 oaordi 8584 nnaordi 8656 unbnn2 9333 ackbij1b 10278 ackbij2 10282 cflm 10290 isf32lem2 10394 indpi 10947 dfon2lem3 35786 |
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