Theorem dfoddfin2 4513

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

Assertion

Ref	Expression
dfoddfin2	|- Oddfin = {x | E.n e. Nn (x = ((n +o n) +o 1c) /\ ((n +o n) +o 1c) =/= (/))}

Distinct variable group:   x,n

Proof of Theorem dfoddfin2

Step	Hyp	Ref	Expression
1		df-oddfin 4445	. 2 |- Oddfin = {x | (E.n e. Nn x = ((n +o n) +o 1c) /\ x =/= (/))}
2		r19.41v 2764	. . . 4 |- (E.n e. Nn (x = ((n +o n) +o 1c) /\ x =/= (/)) <-> (E.n e. Nn x = ((n +o n) +o 1c) /\ x =/= (/)))
3		neeq1 2524	. . . . . 6 |- (x = ((n +o n) +o 1c) -> (x =/= (/) <-> ((n +o n) +o 1c) =/= (/)))
4	3	pm5.32i 618	. . . . 5 |- ((x = ((n +o n) +o 1c) /\ x =/= (/)) <-> (x = ((n +o n) +o 1c) /\ ((n +o n) +o 1c) =/= (/)))
5	4	rexbii 2639	. . . 4 |- (E.n e. Nn (x = ((n +o n) +o 1c) /\ x =/= (/)) <-> E.n e. Nn (x = ((n +o n) +o 1c) /\ ((n +o n) +o 1c) =/= (/)))
6	2, 5	bitr3i 242	. . 3 |- ((E.n e. Nn x = ((n +o n) +o 1c) /\ x =/= (/)) <-> E.n e. Nn (x = ((n +o n) +o 1c) /\ ((n +o n) +o 1c) =/= (/)))
7	6	abbii 2465	. 2 |- {x | (E.n e. Nn x = ((n +o n) +o 1c) /\ x =/= (/))} = {x | E.n e. Nn (x = ((n +o n) +o 1c) /\ ((n +o n) +o 1c) =/= (/))}
8	1, 7	eqtri 2373	1 |- Oddfin = {x | E.n e. Nn (x = ((n +o n) +o 1c) /\ ((n +o n) +o 1c) =/= (/))}

Syntax hints:   /\ wa 358   = wceq 1642  {cab 2339   =/= wne 2516  E.wrex 2615  (/)c0 3550  1_cc1c 4134   Nn cnnc 4373   +o cplc 4375   Odd_fin coddfin 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 2518  df-rex 2620  df-oddfin 4445

This theorem is referenced by:  evenodddisj  4516