Theorem orddif

Description: Ordinal derived from its successor. (Contributed by NM, 20-May-1998)

	Ref	Expression
Assertion	orddif	⊢ ( Ord 𝐴 → 𝐴 = ( suc 𝐴 ∖ { 𝐴 } ) )

Proof

Step	Hyp	Ref	Expression
1		orddisj	⊢ ( Ord 𝐴 → ( 𝐴 ∩ { 𝐴 } ) = ∅ )
2		disj3	⊢ ( ( 𝐴 ∩ { 𝐴 } ) = ∅ ↔ 𝐴 = ( 𝐴 ∖ { 𝐴 } ) )
3		df-suc	⊢ suc 𝐴 = ( 𝐴 ∪ { 𝐴 } )
4	3	difeq1i	⊢ ( suc 𝐴 ∖ { 𝐴 } ) = ( ( 𝐴 ∪ { 𝐴 } ) ∖ { 𝐴 } )
5		difun2	⊢ ( ( 𝐴 ∪ { 𝐴 } ) ∖ { 𝐴 } ) = ( 𝐴 ∖ { 𝐴 } )
6	4 5	eqtri	⊢ ( suc 𝐴 ∖ { 𝐴 } ) = ( 𝐴 ∖ { 𝐴 } )
7	6	eqeq2i	⊢ ( 𝐴 = ( suc 𝐴 ∖ { 𝐴 } ) ↔ 𝐴 = ( 𝐴 ∖ { 𝐴 } ) )
8	2 7	bitr4i	⊢ ( ( 𝐴 ∩ { 𝐴 } ) = ∅ ↔ 𝐴 = ( suc 𝐴 ∖ { 𝐴 } ) )
9	1 8	sylib	⊢ ( Ord 𝐴 → 𝐴 = ( suc 𝐴 ∖ { 𝐴 } ) )