pncan1 - Intuitionistic Logic Explorer

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Theorem pncan1 8667

Description: Cancellation law for addition and subtraction with 1. (Contributed by Alexander van der Vekens, 3-Oct-2018.)

**Assertion**
Ref	Expression
pncan1

**Proof of Theorem pncan1**
Step	Hyp	Ref	Expression
1		id 19	. 2
2		ax-1cn 8236	. . 3
3	2	a1i 9	. 2
4	1, 3	pncand 8601	1

Colors of variables: wff set class

Syntax hints:

wi 4

wceq 1398

wcel 2205 (class class class)co 6058

cc 8141

c1 8144

caddc 8146

cmin 8460

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-14 2208 ax-ext 2216 ax-sep 4233 ax-pow 4292 ax-pr 4327 ax-setind 4664 ax-resscn 8235 ax-1cn 8236 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-addcom 8243 ax-addass 8245 ax-distr 8247 ax-i2m1 8248 ax-0id 8251 ax-rnegex 8252 ax-cnre 8254

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-ral 2527 df-rex 2528 df-reu 2529 df-rab 2531 df-v 2817 df-sbc 3046 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-br 4115 df-opab 4177 df-id 4419 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-iota 5317 df-fun 5359 df-fv 5365 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-sub 8462

This theorem is referenced by: nn0split 10492 nn0disj 10494 seq3f1olemqsumkj 10897 sqoddm1div8 11080 wrdlenccats1lenm1g 11349 ccats1pfxeq 11431 resqrexlemcalc3 11726 ltoddhalfle 12604 flodddiv4 12647 2lgslem1c 16089 2lgslem3a 16092 wlklenvm1 16462 wlklenvm1g 16463 depindlem1 16627