|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > elsuc | Structured version Visualization version GIF version | ||
| Description: Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-2003.) | 
| Ref | Expression | 
|---|---|
| elsuc.1 | ⊢ 𝐴 ∈ V | 
| Ref | Expression | 
|---|---|
| elsuc | ⊢ (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elsuc.1 | . 2 ⊢ 𝐴 ∈ V | |
| 2 | elsucg 6451 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∨ wo 847 = wceq 1539 ∈ wcel 2107 Vcvv 3479 suc csuc 6385 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-un 3955 df-sn 4626 df-suc 6389 | 
| This theorem is referenced by: sucel 6457 limsssuc 7872 omsmolem 8696 cantnfle 9712 infxpenlem 10054 inatsk 10819 nolesgn2ores 27718 nogesgn1ores 27720 untsucf 35711 dfon2lem7 35791 rdgssun 37380 omssaxinf2 45010 | 
| Copyright terms: Public domain | W3C validator |