Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > nfsuc | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for successor. (Contributed by NM, 15-Sep-2003.) |
Ref | Expression |
---|---|
nfsuc.1 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
nfsuc | ⊢ Ⅎ𝑥 suc 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-suc 6197 | . 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
2 | nfsuc.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | nfsn 4643 | . . 3 ⊢ Ⅎ𝑥{𝐴} |
4 | 2, 3 | nfun 4141 | . 2 ⊢ Ⅎ𝑥(𝐴 ∪ {𝐴}) |
5 | 1, 4 | nfcxfr 2975 | 1 ⊢ Ⅎ𝑥 suc 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnfc 2961 ∪ cun 3934 {csn 4567 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 |
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-v 3496 df-un 3941 df-sn 4568 df-pr 4570 df-suc 6197 |
This theorem is referenced by: rankidb 9229 dfon2lem3 33030 |
Copyright terms: Public domain | W3C validator |