Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > elsuc2 | Structured version Visualization version GIF version |
Description: Membership in a successor. (Contributed by NM, 15-Sep-2003.) |
Ref | Expression |
---|---|
elsuc.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
elsuc2 | ⊢ (𝐵 ∈ suc 𝐴 ↔ (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elsuc.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | elsuc2g 5793 | . 2 ⊢ (𝐴 ∈ V → (𝐵 ∈ suc 𝐴 ↔ (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐵 ∈ suc 𝐴 ↔ (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∨ wo 383 = wceq 1483 ∈ wcel 1990 Vcvv 3200 suc csuc 5725 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-un 3579 df-sn 4178 df-suc 5729 |
This theorem is referenced by: alephordi 8897 |
Copyright terms: Public domain | W3C validator |