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Mirrors > Home > MPE Home > Th. List > sucexg | Structured version Visualization version GIF version |
Description: The successor of a set is a set (generalization). (Contributed by NM, 5-Jun-1994.) |
Ref | Expression |
---|---|
sucexg | ⊢ (𝐴 ∈ 𝑉 → suc 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3459 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | sucexb 7504 | . 2 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V) | |
3 | 1, 2 | sylib 221 | 1 ⊢ (𝐴 ∈ 𝑉 → suc 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 Vcvv 3441 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 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-sn 4526 df-pr 4528 df-uni 4801 df-suc 6165 |
This theorem is referenced by: sucex 7506 suceloni 7508 hsmexlem1 9837 dfon2lem3 33143 inaex 41005 |
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