Theorem sucidALTVD

Description: A set belongs to its successor. Alternate proof of sucid . The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sucidALT is sucidALTVD without virtual deductions and was automatically derived from sucidALTVD . This proof illustrates that completeusersproof.cmd will generate a Metamath proof from any User's Proof which is "conventional" in the sense that no step is a virtual deduction, provided that all necessary unification theorems and transformation deductions are in set.mm. completeusersproof.cmd automatically converts such a conventional proof into a Virtual Deduction proof for which each step happens to be a 0-virtual hypothesis virtual deduction. The user does not need to search for reference theorem labels or deduction labels nor does he(she) need to use theorems and deductions which unify with reference theorems and deductions in set.mm. All that is necessary is that each theorem or deduction of the User's Proof unifies with some reference theorem or deduction in set.mm or is a semantic variation of some theorem or deduction which unifies with some reference theorem or deduction in set.mm. The definition of "semantic variation" has not been precisely defined. If it is obvious that a theorem or deduction has the same meaning as another theorem or deduction, then it is a semantic variation of the latter theorem or deduction. For example, step 4 of the User's Proof is a semantic variation of the definition (axiom) suc A = ( A u. { A } ) , which unifies with df-suc , a reference definition (axiom) in set.mm. Also, a theorem or deduction is said to be a semantic variation of another theorem or deduction if it is obvious upon cursory inspection that it has the same meaning as a weaker form of the latter theorem or deduction. For example, the deduction Ord A infers A. x e. A A. y e. A ( x e. y \/ x = y \/ y e. x ) is a semantic variation of the theorem ( Ord A <-> ( Tr A /\ A. x e. A A. y e. A ( x e. y \/ x = y \/ y e. x ) ) ) , which unifies with the set.mm reference definition (axiom) dford2 .

	Ref	Expression
Hypothesis	sucidALTVD.1	⊢ 𝐴 ∈ V
Assertion	sucidALTVD	⊢ 𝐴 ∈ suc 𝐴

Proof

Step	Hyp	Ref	Expression
1		sucidALTVD.1	⊢ 𝐴 ∈ V
2	1	snid	⊢ 𝐴 ∈ { 𝐴 }
3		elun1	⊢ ( 𝐴 ∈ { 𝐴 } → 𝐴 ∈ ( { 𝐴 } ∪ 𝐴 ) )
4	2 3	e0a	⊢ 𝐴 ∈ ( { 𝐴 } ∪ 𝐴 )
5		df-suc	⊢ suc 𝐴 = ( 𝐴 ∪ { 𝐴 } )
6	5	equncomi	⊢ suc 𝐴 = ( { 𝐴 } ∪ 𝐴 )
7	4 6	eleqtrri	⊢ 𝐴 ∈ suc 𝐴