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Theorem hashun2 13172

Description: The size of the union of finite sets is less than or equal to the sum of their sizes. (Contributed by Mario Carneiro, 23-Sep-2013.) (Proof shortened by Mario Carneiro, 27-Jul-2014.)

**Assertion**
Ref	Expression
hashun2	⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (#‘(𝐴 ∪ 𝐵)) ≤ ((#‘𝐴) + (#‘𝐵)))

**Proof of Theorem hashun2**
Step	Hyp	Ref	Expression
1		undif2 4044	. . . 4 ⊢ (𝐴 ∪ (𝐵 ∖ 𝐴)) = (𝐴 ∪ 𝐵)
2	1	fveq2i 6194	. . 3 ⊢ (#‘(𝐴 ∪ (𝐵 ∖ 𝐴))) = (#‘(𝐴 ∪ 𝐵))
3		diffi 8192	. . . 4 ⊢ (𝐵 ∈ Fin → (𝐵 ∖ 𝐴) ∈ Fin)
4		disjdif 4040	. . . . 5 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅
5		hashun 13171	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ (𝐵 ∖ 𝐴) ∈ Fin ∧ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅) → (#‘(𝐴 ∪ (𝐵 ∖ 𝐴))) = ((#‘𝐴) + (#‘(𝐵 ∖ 𝐴))))
6	4, 5	mp3an3 1413	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ (𝐵 ∖ 𝐴) ∈ Fin) → (#‘(𝐴 ∪ (𝐵 ∖ 𝐴))) = ((#‘𝐴) + (#‘(𝐵 ∖ 𝐴))))
7	3, 6	sylan2 491	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (#‘(𝐴 ∪ (𝐵 ∖ 𝐴))) = ((#‘𝐴) + (#‘(𝐵 ∖ 𝐴))))
8	2, 7	syl5eqr 2670	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (#‘(𝐴 ∪ 𝐵)) = ((#‘𝐴) + (#‘(𝐵 ∖ 𝐴))))
9	3	adantl 482	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐵 ∖ 𝐴) ∈ Fin)
10		hashcl 13147	. . . . 5 ⊢ ((𝐵 ∖ 𝐴) ∈ Fin → (#‘(𝐵 ∖ 𝐴)) ∈ ℕ₀)
11	9, 10	syl 17	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (#‘(𝐵 ∖ 𝐴)) ∈ ℕ₀)
12	11	nn0red 11352	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (#‘(𝐵 ∖ 𝐴)) ∈ ℝ)
13		hashcl 13147	. . . . 5 ⊢ (𝐵 ∈ Fin → (#‘𝐵) ∈ ℕ₀)
14	13	adantl 482	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (#‘𝐵) ∈ ℕ₀)
15	14	nn0red 11352	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (#‘𝐵) ∈ ℝ)
16		hashcl 13147	. . . . 5 ⊢ (𝐴 ∈ Fin → (#‘𝐴) ∈ ℕ₀)
17	16	adantr 481	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (#‘𝐴) ∈ ℕ₀)
18	17	nn0red 11352	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (#‘𝐴) ∈ ℝ)
19		simpr 477	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → 𝐵 ∈ Fin)
20		difss 3737	. . . . 5 ⊢ (𝐵 ∖ 𝐴) ⊆ 𝐵
21		ssdomg 8001	. . . . 5 ⊢ (𝐵 ∈ Fin → ((𝐵 ∖ 𝐴) ⊆ 𝐵 → (𝐵 ∖ 𝐴) ≼ 𝐵))
22	19, 20, 21	mpisyl 21	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐵 ∖ 𝐴) ≼ 𝐵)
23		hashdom 13168	. . . . 5 ⊢ (((𝐵 ∖ 𝐴) ∈ Fin ∧ 𝐵 ∈ Fin) → ((#‘(𝐵 ∖ 𝐴)) ≤ (#‘𝐵) ↔ (𝐵 ∖ 𝐴) ≼ 𝐵))
24	9, 23	sylancom 701	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((#‘(𝐵 ∖ 𝐴)) ≤ (#‘𝐵) ↔ (𝐵 ∖ 𝐴) ≼ 𝐵))
25	22, 24	mpbird 247	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (#‘(𝐵 ∖ 𝐴)) ≤ (#‘𝐵))
26	12, 15, 18, 25	leadd2dd 10642	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((#‘𝐴) + (#‘(𝐵 ∖ 𝐴))) ≤ ((#‘𝐴) + (#‘𝐵)))
27	8, 26	eqbrtrd 4675	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (#‘(𝐴 ∪ 𝐵)) ≤ ((#‘𝐴) + (#‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∖ cdif 3571 ∪ cun 3572 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ≼ cdom 7953 Fincfn 7955 + caddc 9939 ≤ cle 10075 ℕ₀cn0 11292 #chash 13117

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-fz 12327 df-hash 13118

This theorem is referenced by: hashunlei 13212 hashfun 13224 prmreclem4 15623 fta1glem2 23926 fta1lem 24062 vieta1lem2 24066

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