nnncan2 - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > nnncan2		Unicode version

Theorem nnncan2 8208

Description: Cancellation law for subtraction. (Contributed by NM, 1-Oct-2005.)

**Assertion**
Ref	Expression
nnncan2	$/\$ $/\$

**Proof of Theorem nnncan2**
Step	Hyp	Ref	Expression
1		subcl 8170	. . . 4 $/\$
2	1	3adant1 1016	. . 3 $/\$ $/\$
3		sub32 8205	. . 3 $/\$ $/\$
4	2, 3	syld3an2 1295	. 2 $/\$ $/\$
5		nnncan 8206	. 2 $/\$ $/\$
6	4, 5	eqtr3d 2222	1 $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ w3a 979

wceq 1363

wcel 2158 (class class class)co 5888

cc 7823

cmin 8142

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-setind 4548 ax-resscn 7917 ax-1cn 7918 ax-icn 7920 ax-addcl 7921 ax-addrcl 7922 ax-mulcl 7923 ax-addcom 7925 ax-addass 7927 ax-distr 7929 ax-i2m1 7930 ax-0id 7933 ax-rnegex 7934 ax-cnre 7936

This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-opab 4077 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-iota 5190 df-fun 5230 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-sub 8144

This theorem is referenced by: nnncan2d 8317 fzmmmeqm 10072