Theorem zfpair 1891

Description: The Pairing axiom 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 can be derived from the Extensionality, Replacement, and Power Set axioms.

Assertion

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

Proof of Theorem zfpair

Step	Hyp	Ref	Expression
1		dfpr2 1821	. 2 |- {x, y} = {w | (w = x \/ w = y)}
2		19.43 767	. . . . 5 |- (E.z((z = (/) /\ w = x) \/ (z = {(/)} /\ w = y)) <-> (E.z(z = (/) /\ w = x) \/ E.z(z = {(/)} /\ w = y)))
3		prlem2 577	. . . . . 6 |- (((z = (/) /\ w = x) \/ (z = {(/)} /\ w = y)) <-> ((z = (/) \/ z = {(/)}) /\ ((z = (/) /\ w = x) \/ (z = {(/)} /\ w = y))))
4	3	biex 733	. . . . 5 |- (E.z((z = (/) /\ w = x) \/ (z = {(/)} /\ w = y)) <-> E.z((z = (/) \/ z = {(/)}) /\ ((z = (/) /\ w = x) \/ (z = {(/)} /\ w = y))))
5		19.41v 963	. . . . . . 7 |- (E.z(z = (/) /\ w = x) <-> (E.z z = (/) /\ w = x))
6		0ex 1745	. . . . . . . 8 |- (/) e. V
7	6	isseti 1352	. . . . . . 7 |- E.z z = (/)
8	5, 7	mpbiran 547	. . . . . 6 |- (E.z(z = (/) /\ w = x) <-> w = x)
9		19.41v 963	. . . . . . 7 |- (E.z(z = {(/)} /\ w = y) <-> (E.z z = {(/)} /\ w = y))
10		p0ex 1885	. . . . . . . 8 |- {(/)} e. V
11	10	isseti 1352	. . . . . . 7 |- E.z z = {(/)}
12	9, 11	mpbiran 547	. . . . . 6 |- (E.z(z = {(/)} /\ w = y) <-> w = y)
13	8, 12	orbi12i 216	. . . . 5 |- ((E.z(z = (/) /\ w = x) \/ E.z(z = {(/)} /\ w = y)) <-> (w = x \/ w = y))
14	2, 4, 13	3bitr3r 157	. . . 4 |- ((w = x \/ w = y) <-> E.z((z = (/) \/ z = {(/)}) /\ ((z = (/) /\ w = x) \/ (z = {(/)} /\ w = y))))
15	14	biabi 1181	. . 3 |- {w | (w = x \/ w = y)} = {w | E.z((z = (/) \/ z = {(/)}) /\ ((z = (/) /\ w = x) \/ (z = {(/)} /\ w = y)))}
16		dfpr2 1821	. . . . 5 |- {(/), {(/)}} = {z | (z = (/) \/ z = {(/)})}
17		pp0ex 1886	. . . . 5 |- {(/), {(/)}} e. V
18	16, 17	eqeltrr 1160	. . . 4 |- {z | (z = (/) \/ z = {(/)})} e. V
19		eqt2b 818	. . . . . . . 8 |- (v = x -> (w = v <-> w = x))
20		0inp0 1888	. . . . . . . 8 |- (z = (/) -> -. z = {(/)})
21	19, 20	prlem1 576	. . . . . . 7 |- (v = x -> (z = (/) -> (((z = (/) /\ w = x) \/ (z = {(/)} /\ w = y)) -> w = v)))
22	21	19.21adv 945	. . . . . 6 |- (v = x -> (z = (/) -> A.w(((z = (/) /\ w = x) \/ (z = {(/)} /\ w = y)) -> w = v)))
23	22	a4w 929	. . . . 5 |- (z = (/) -> E.vA.w(((z = (/) /\ w = x) \/ (z = {(/)} /\ w = y)) -> w = v))
24		eqt2b 818	. . . . . . . . 9 |- (v = y -> (w = v <-> w = y))
25	20	con2i 89	. . . . . . . . 9 |- (z = {(/)} -> -. z = (/))
26	24, 25	prlem1 576	. . . . . . . 8 |- (v = y -> (z = {(/)} -> (((z = {(/)} /\ w = y) \/ (z = (/) /\ w = x)) -> w = v)))
27		orcom 209	. . . . . . . 8 |- (((z = (/) /\ w = x) \/ (z = {(/)} /\ w = y)) <-> ((z = {(/)} /\ w = y) \/ (z = (/) /\ w = x)))
28	26, 27	bisyl7 189	. . . . . . 7 |- (v = y -> (z = {(/)} -> (((z = (/) /\ w = x) \/ (z = {(/)} /\ w = y)) -> w = v)))
29	28	19.21adv 945	. . . . . 6 |- (v = y -> (z = {(/)} -> A.w(((z = (/) /\ w = x) \/ (z = {(/)} /\ w = y)) -> w = v)))
30	29	a4w 929	. . . . 5 |- (z = {(/)} -> E.vA.w(((z = (/) /\ w = x) \/ (z = {(/)} /\ w = y)) -> w = v))
31	23, 30	jaoi 275	. . . 4 |- ((z = (/) \/ z = {(/)}) -> E.vA.w(((z = (/) /\ w = x) \/ (z = {(/)} /\ w = y)) -> w = v))
32	18, 31	zfrep4 1479	. . 3 |- {w | E.z((z = (/) \/ z = {(/)}) /\ ((z = (/) /\ w = x) \/ (z = {(/)} /\ w = y)))} e. V
33	15, 32	eqeltr 1159	. 2 |- {w | (w = x \/ w = y)} e. V
34	1, 33	eqeltr 1159	1 |- {x, y} e. V

Syntax hints:   -> wi 2   \/ wo 195   /\ wa 196  A.wal 672  E.wex 678   = weq 797  {cab 1090   = wceq 1091   e. wcel 1092  Vcvv 1348  (/)c0 1707  {csn 1808  {cpr 1809

This theorem is referenced by:  prex 1892  pwssun 1917  fr2nr 2177  xpex 2488  fiint 3445

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812

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