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Theorem opkthg 4131

Description: Two Kuratowski ordered pairs are equal iff their components are equal. (Contributed by SF, 12-Jan-2015.)

**Assertion**
Ref	Expression
opkthg	$/\$ $/\$ $/\$

**Proof of Theorem opkthg**
Step	Hyp	Ref	Expression
1		simp1 955	. . . . 5 $/\$ $/\$
2		opkth1g 4130	. . . . 5 $/\$
3	1, 2	sylan 457	. . . 4 $/\$ $/\$ $/\$
4		simp2 956	. . . . . 6 $/\$ $/\$
5		simp3 957	. . . . . 6 $/\$ $/\$
6	4, 5	jca 518	. . . . 5 $/\$ $/\$ $/\$
7		opkeq1 4059	. . . . . . . . . . 11
8	7	eqeq1d 2361	. . . . . . . . . 10
9	8	biimpd 198	. . . . . . . . 9
10	9	impcom 419	. . . . . . . 8 $/\$
11		df-opk 4058	. . . . . . . . . . 11
12		df-opk 4058	. . . . . . . . . . 11
13	11, 12	eqeq12i 2366	. . . . . . . . . 10
14		prex 4112	. . . . . . . . . . 11
15		prex 4112	. . . . . . . . . . 11
16	14, 15	preqr2 4125	. . . . . . . . . 10
17	13, 16	sylbi 187	. . . . . . . . 9
18		preqr2g 4126	. . . . . . . . 9 $/\$
19	17, 18	syl5 28	. . . . . . . 8 $/\$
20	10, 19	syl5 28	. . . . . . 7 $/\$ $/\$
21	20	exp3a 425	. . . . . 6 $/\$
22	21	imp 418	. . . . 5 $/\$ $/\$
23	6, 22	sylan 457	. . . 4 $/\$ $/\$ $/\$
24	3, 23	jcai 522	. . 3 $/\$ $/\$ $/\$ $/\$
25	24	ex 423	. 2 $/\$ $/\$ $/\$
26		opkeq12 4061	. 2 $/\$
27	25, 26	impbid1 194	1 $/\$ $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 176 $/\$ wa 358 $/\$ w3a 934

wceq 1642

wcel 1710

csn 3737

cpr 3738

copk 4057

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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 ax-nin 4078 ax-sn 4087

This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-ss 3259 df-nul 3551 df-sn 3741 df-pr 3742 df-opk 4058

This theorem is referenced by: opkth 4132 opkelopkabg 4245 otkelins2kg 4253 otkelins3kg 4254 opkelcokg 4261