Theorem sucidALTVD 39106

Description: A set belongs to its successor. Alternate proof of sucid 5804. 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 39107 is sucidALTVD 39106 without virtual deductions and was automatically derived from sucidALTVD 39106. 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 5729, 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 8517.

h1::	|- A e. _V
2:1:	|- A e. { A }
3:2:	|- A e. ( { A } u. A )
4::	|- suc A = ( { A } u. A )
qed:3,4:	|- A e. suc A

(Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
sucidALTVD.1	|- A e. _V

Assertion

Ref	Expression
sucidALTVD	|- A e. suc A

Proof of Theorem sucidALTVD

Step	Hyp	Ref	Expression
1		sucidALTVD.1	. . . 4 |- A e. _V
2	1	snid 4208	. . 3 |- A e. { A }
3		elun1 3780	. . 3 |- ( A e. { A } -> A e. ( { A } u. A ) )
4	2, 3	e0a 38999	. 2 |- A e. ( { A } u. A )
5		df-suc 5729	. . 3 |- suc A = ( A u. { A } )
6	5	equncomi 3759	. 2 |- suc A = ( { A } u. A )
7	4, 6	eleqtrri 2700	1 |- A e. suc A

Syntax hints:   e. wcel 1990   _Vcvv 3200   u. cun 3572   {csn 4177   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-in 3581  df-ss 3588  df-sn 4178  df-suc 5729

This theorem is referenced by: (None)