Theorem zfpair 4904

Description: The Axiom of Pairing of Zermelo-Fraenkel set theory. Axiom 2 of [TakeutiZaring] p. 15. In some textbooks this is stated as a separate axiom; here we show it is redundant since it can be derived from the other axioms.

This theorem should not be referenced by any proof other than axpr 4905. Instead, use zfpair2 4907 below so that the uses of the Axiom of Pairing can be more easily identified. (Contributed by NM, 18-Oct-1995.) (New usage is discouraged.)

Assertion

Ref	Expression
zfpair	|- { x , y } e. _V

Proof of Theorem zfpair

Dummy variables z w v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfpr2 4195	. 2 |- { x , y } = { w | ( w = x \/ w = y ) }
2		19.43 1810	. . . . 5 |- ( E. z ( ( z = (/) /\ w = x ) \/ ( z = { (/) } /\ w = y ) ) <-> ( E. z ( z = (/) /\ w = x ) \/ E. z ( z = { (/) } /\ w = y ) ) )
3		prlem2 1006	. . . . . 6 |- ( ( ( z = (/) /\ w = x ) \/ ( z = { (/) } /\ w = y ) ) <-> ( ( z = (/) \/ z = { (/) } ) /\ ( ( z = (/) /\ w = x ) \/ ( z = { (/) } /\ w = y ) ) ) )
4	3	exbii 1774	. . . . 5 |- ( E. z ( ( z = (/) /\ w = x ) \/ ( z = { (/) } /\ w = y ) ) <-> E. z ( ( z = (/) \/ z = { (/) } ) /\ ( ( z = (/) /\ w = x ) \/ ( z = { (/) } /\ w = y ) ) ) )
5		0ex 4790	. . . . . . . 8 |- (/) e. _V
6	5	isseti 3209	. . . . . . 7 |- E. z z = (/)
7		19.41v 1914	. . . . . . 7 |- ( E. z ( z = (/) /\ w = x ) <-> ( E. z z = (/) /\ w = x ) )
8	6, 7	mpbiran 953	. . . . . 6 |- ( E. z ( z = (/) /\ w = x ) <-> w = x )
9		p0ex 4853	. . . . . . . 8 |- { (/) } e. _V
10	9	isseti 3209	. . . . . . 7 |- E. z z = { (/) }
11		19.41v 1914	. . . . . . 7 |- ( E. z ( z = { (/) } /\ w = y ) <-> ( E. z z = { (/) } /\ w = y ) )
12	10, 11	mpbiran 953	. . . . . 6 |- ( E. z ( z = { (/) } /\ w = y ) <-> w = y )
13	8, 12	orbi12i 543	. . . . 5 |- ( ( E. z ( z = (/) /\ w = x ) \/ E. z ( z = { (/) } /\ w = y ) ) <-> ( w = x \/ w = y ) )
14	2, 4, 13	3bitr3ri 291	. . . 4 |- ( ( w = x \/ w = y ) <-> E. z ( ( z = (/) \/ z = { (/) } ) /\ ( ( z = (/) /\ w = x ) \/ ( z = { (/) } /\ w = y ) ) ) )
15	14	abbii 2739	. . 3 |- { w | ( w = x \/ w = y ) } = { w | E. z ( ( z = (/) \/ z = { (/) } ) /\ ( ( z = (/) /\ w = x ) \/ ( z = { (/) } /\ w = y ) ) ) }
16		dfpr2 4195	. . . . 5 |- { (/) , { (/) } } = { z | ( z = (/) \/ z = { (/) } ) }
17		pp0ex 4855	. . . . 5 |- { (/) , { (/) } } e. _V
18	16, 17	eqeltrri 2698	. . . 4 |- { z | ( z = (/) \/ z = { (/) } ) } e. _V
19		equequ2 1953	. . . . . . . 8 |- ( v = x -> ( w = v <-> w = x ) )
20		0inp0 4837	. . . . . . . 8 |- ( z = (/) -> -. z = { (/) } )
21	19, 20	prlem1 1005	. . . . . . 7 |- ( v = x -> ( z = (/) -> ( ( ( z = (/) /\ w = x ) \/ ( z = { (/) } /\ w = y ) ) -> w = v ) ) )
22	21	alrimdv 1857	. . . . . 6 |- ( v = x -> ( z = (/) -> A. w ( ( ( z = (/) /\ w = x ) \/ ( z = { (/) } /\ w = y ) ) -> w = v ) ) )
23	22	spimev 2259	. . . . 5 |- ( z = (/) -> E. v A. w ( ( ( z = (/) /\ w = x ) \/ ( z = { (/) } /\ w = y ) ) -> w = v ) )
24		orcom 402	. . . . . . . 8 |- ( ( ( z = (/) /\ w = x ) \/ ( z = { (/) } /\ w = y ) ) <-> ( ( z = { (/) } /\ w = y ) \/ ( z = (/) /\ w = x ) ) )
25		equequ2 1953	. . . . . . . . 9 |- ( v = y -> ( w = v <-> w = y ) )
26	20	con2i 134	. . . . . . . . 9 |- ( z = { (/) } -> -. z = (/) )
27	25, 26	prlem1 1005	. . . . . . . 8 |- ( v = y -> ( z = { (/) } -> ( ( ( z = { (/) } /\ w = y ) \/ ( z = (/) /\ w = x ) ) -> w = v ) ) )
28	24, 27	syl7bi 245	. . . . . . 7 |- ( v = y -> ( z = { (/) } -> ( ( ( z = (/) /\ w = x ) \/ ( z = { (/) } /\ w = y ) ) -> w = v ) ) )
29	28	alrimdv 1857	. . . . . 6 |- ( v = y -> ( z = { (/) } -> A. w ( ( ( z = (/) /\ w = x ) \/ ( z = { (/) } /\ w = y ) ) -> w = v ) ) )
30	29	spimev 2259	. . . . 5 |- ( z = { (/) } -> E. v A. w ( ( ( z = (/) /\ w = x ) \/ ( z = { (/) } /\ w = y ) ) -> w = v ) )
31	23, 30	jaoi 394	. . . 4 |- ( ( z = (/) \/ z = { (/) } ) -> E. v A. w ( ( ( z = (/) /\ w = x ) \/ ( z = { (/) } /\ w = y ) ) -> w = v ) )
32	18, 31	zfrep4 4779	. . 3 |- { w | E. z ( ( z = (/) \/ z = { (/) } ) /\ ( ( z = (/) /\ w = x ) \/ ( z = { (/) } /\ w = y ) ) ) } e. _V
33	15, 32	eqeltri 2697	. 2 |- { w | ( w = x \/ w = y ) } e. _V
34	1, 33	eqeltri 2697	1 |- { x , y } e. _V

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   _Vcvv 3200   (/)c0 3915   {csn 4177   {cpr 4179

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  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  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-ne 2795  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-pw 4160  df-sn 4178  df-pr 4180

This theorem is referenced by:  axpr  4905