Theorem dfevenfin2 4513

Description: Alternate definition of even number. (Contributed by SF, 25-Jan-2015.)

Assertion

Ref	Expression
dfevenfin2	⊢ Evenfin = {x ∣ ∃n ∈ Nn (x = (n +c n) ∧ (n +c n) ≠ ∅)}

Distinct variable group:   x,n

Proof of Theorem dfevenfin2

Step	Hyp	Ref	Expression
1		df-evenfin 4445	. 2 ⊢ Evenfin = {x ∣ (∃n ∈ Nn x = (n +c n) ∧ x ≠ ∅)}
2		r19.41v 2765	. . . 4 ⊢ (∃n ∈ Nn (x = (n +c n) ∧ x ≠ ∅) ↔ (∃n ∈ Nn x = (n +c n) ∧ x ≠ ∅))
3		neeq1 2525	. . . . . 6 ⊢ (x = (n +c n) → (x ≠ ∅ ↔ (n +c n) ≠ ∅))
4	3	pm5.32i 618	. . . . 5 ⊢ ((x = (n +c n) ∧ x ≠ ∅) ↔ (x = (n +c n) ∧ (n +c n) ≠ ∅))
5	4	rexbii 2640	. . . 4 ⊢ (∃n ∈ Nn (x = (n +c n) ∧ x ≠ ∅) ↔ ∃n ∈ Nn (x = (n +c n) ∧ (n +c n) ≠ ∅))
6	2, 5	bitr3i 242	. . 3 ⊢ ((∃n ∈ Nn x = (n +c n) ∧ x ≠ ∅) ↔ ∃n ∈ Nn (x = (n +c n) ∧ (n +c n) ≠ ∅))
7	6	abbii 2466	. 2 ⊢ {x ∣ (∃n ∈ Nn x = (n +c n) ∧ x ≠ ∅)} = {x ∣ ∃n ∈ Nn (x = (n +c n) ∧ (n +c n) ≠ ∅)}
8	1, 7	eqtri 2373	1 ⊢ Evenfin = {x ∣ ∃n ∈ Nn (x = (n +c n) ∧ (n +c n) ≠ ∅)}

Syntax hints:   ∧ wa 358   = wceq 1642  {cab 2339   ≠ wne 2517  ∃wrex 2616  ∅c0 3551   Nn cnnc 4374   +_c cplc 4376   Even_fin cevenfin 4437

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-ne 2519  df-rex 2621  df-evenfin 4445

This theorem is referenced by:  evenodddisj  4517