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Theorem nfunv 5138

Description: The universe is not a function. (Contributed by Raph Levien, 27-Jan-2004.)

**Assertion**
Ref	Expression
nfunv

Proof of Theorem nfunv Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2862	. . . . . . 7
2		vex 2862	. . . . . . 7
3	1, 2	opex 4588	. . . . . 6
4	2	complex 4104	. . . . . . 7 ∼
5	1, 4	opex 4588	. . . . . 6 ∼
6	3, 5	pm3.2i 441	. . . . 5 $/\$ ∼
7		necompl 3544	. . . . . 6 ∼
8	7	necomi 2598	. . . . 5 ∼
9	6, 8	pm3.2i 441	. . . 4 $/\$ ∼ $/\$ ∼
10		opeq2 4579	. . . . . . . 8 ∼ ∼
11	10	eleq1d 2419	. . . . . . 7 ∼ ∼
12	11	anbi2d 684	. . . . . 6 ∼ $/\$ $/\$ ∼
13		df-ne 2518	. . . . . . 7
14		neeq2 2525	. . . . . . 7 ∼ ∼
15	13, 14	syl5bbr 250	. . . . . 6 ∼ ∼
16	12, 15	anbi12d 691	. . . . 5 ∼ $/\$ $/\$ $/\$ ∼ $/\$ ∼
17	4, 16	spcev 2946	. . . 4 $/\$ ∼ $/\$ ∼ $/\$ $/\$
18		19.8a 1756	. . . . 5 $/\$ $/\$ $/\$ $/\$
19	18	19.23bi 1759	. . . 4 $/\$ $/\$ $/\$ $/\$
20	9, 17, 19	mp2b 9	. . 3 $/\$ $/\$
21		exanali 1585	. . . . . . 7 $/\$ $/\$ $/\$
22	21	exbii 1582	. . . . . 6 $/\$ $/\$ $/\$
23		exnal 1574	. . . . . 6 $/\$ $/\$
24	22, 23	bitri 240	. . . . 5 $/\$ $/\$ $/\$
25	24	exbii 1582	. . . 4 $/\$ $/\$ $/\$
26		exnal 1574	. . . 4 $/\$ $/\$
27	25, 26	bitri 240	. . 3 $/\$ $/\$ $/\$
28	20, 27	mpbi 199	. 2 $/\$
29		dffun4 5121	. 2 $/\$
30	28, 29	mtbir 290	1

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4 $/\$ wa 358

wal 1540

wex 1541

wceq 1642

wcel 1710

wne 2516

cvv 2859 ∼ ccompl 3205

cop 4561

wfun 4775

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-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087

This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-co 4726 df-id 4767 df-cnv 4785 df-fun 4789

This theorem is referenced by: (None)