eladdci - New Foundations Explorer

	New Foundations Explorer		< Previous Next > Nearby theorems
Mirrors > Home > NFE Home > Th. List > eladdci		Unicode version

Theorem eladdci 4399

Description: Inference form of membership in cardinal addition. (Contributed by SF, 26-Jan-2015.)

**Assertion**
Ref	Expression
eladdci	$/\$ $/\$

Proof of Theorem eladdci Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2353	. . . 4
2		ineq1 3450	. . . . . . . 8
3	2	eqeq1d 2361	. . . . . . 7
4		uneq1 3411	. . . . . . . 8
5	4	eqeq2d 2364	. . . . . . 7
6	3, 5	anbi12d 691	. . . . . 6 $/\$ $/\$
7		ineq2 3451	. . . . . . . 8
8	7	eqeq1d 2361	. . . . . . 7
9		uneq2 3412	. . . . . . . 8
10	9	eqeq2d 2364	. . . . . . 7
11	8, 10	anbi12d 691	. . . . . 6 $/\$ $/\$
12	6, 11	rspc2ev 2963	. . . . 5 $/\$ $/\$ $/\$ $/\$
13	12	3expa 1151	. . . 4 $/\$ $/\$ $/\$ $/\$
14	1, 13	mpanr2 665	. . 3 $/\$ $/\$ $/\$
15	14	3impa 1146	. 2 $/\$ $/\$ $/\$
16		eladdc 4398	. 2 $/\$
17	15, 16	sylibr 203	1 $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4 $/\$ wa 358 $/\$ w3a 934

wceq 1642

wcel 1710

wrex 2615

cun 3207

cin 3208

c0 3550

cplc 4375

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

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-ral 2619 df-rex 2620 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-addc 4378

This theorem is referenced by: ncfindi 4475 tfindi 4496 nnadjoinpw 4521 sfinltfin 4535 ncdisjun 6136 tcdi 6164 ce0addcnnul 6179