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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 3512 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | sucexb 7524 | . 2 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V) | |
3 | 1, 2 | sylib 220 | 1 ⊢ (𝐴 ∈ 𝑉 → suc 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 Vcvv 3494 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 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-un 7461 |
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-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-sn 4568 df-pr 4570 df-uni 4839 df-suc 6197 |
This theorem is referenced by: sucex 7526 suceloni 7528 hsmexlem1 9848 dfon2lem3 33030 inaex 40653 |
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