Theorem infdif 10130

Description: The cardinality of an infinite set does not change after subtracting a strictly smaller one. Example in [Enderton] p. 164. (Contributed by NM, 22-Oct-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)

Assertion

Ref	Expression
infdif	⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → (𝐴 ∖ 𝐵) ≈ 𝐴)

Proof of Theorem infdif

Step	Hyp	Ref	Expression
1		simp1 1137	. . 3 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → 𝐴 ∈ dom card)
2		difss 4090	. . 3 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
3		ssdomg 8949	. . 3 ⊢ (𝐴 ∈ dom card → ((𝐴 ∖ 𝐵) ⊆ 𝐴 → (𝐴 ∖ 𝐵) ≼ 𝐴))
4	1, 2, 3	mpisyl 21	. 2 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → (𝐴 ∖ 𝐵) ≼ 𝐴)
5		sdomdom 8929	. . . . . . . . 9 ⊢ (𝐵 ≺ 𝐴 → 𝐵 ≼ 𝐴)
6	5	3ad2ant3 1136	. . . . . . . 8 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → 𝐵 ≼ 𝐴)
7		numdom 9960	. . . . . . . 8 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ dom card)
8	1, 6, 7	syl2anc 585	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → 𝐵 ∈ dom card)
9		unnum 10119	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ∪ 𝐵) ∈ dom card)
10	1, 8, 9	syl2anc 585	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → (𝐴 ∪ 𝐵) ∈ dom card)
11		ssun1 4132	. . . . . 6 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
12		ssdomg 8949	. . . . . 6 ⊢ ((𝐴 ∪ 𝐵) ∈ dom card → (𝐴 ⊆ (𝐴 ∪ 𝐵) → 𝐴 ≼ (𝐴 ∪ 𝐵)))
13	10, 11, 12	mpisyl 21	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → 𝐴 ≼ (𝐴 ∪ 𝐵))
14		undif1 4430	. . . . . 6 ⊢ ((𝐴 ∖ 𝐵) ∪ 𝐵) = (𝐴 ∪ 𝐵)
15		ssnum 9961	. . . . . . . 8 ⊢ ((𝐴 ∈ dom card ∧ (𝐴 ∖ 𝐵) ⊆ 𝐴) → (𝐴 ∖ 𝐵) ∈ dom card)
16	1, 2, 15	sylancl 587	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → (𝐴 ∖ 𝐵) ∈ dom card)
17		undjudom 10090	. . . . . . 7 ⊢ (((𝐴 ∖ 𝐵) ∈ dom card ∧ 𝐵 ∈ dom card) → ((𝐴 ∖ 𝐵) ∪ 𝐵) ≼ ((𝐴 ∖ 𝐵) ⊔ 𝐵))
18	16, 8, 17	syl2anc 585	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → ((𝐴 ∖ 𝐵) ∪ 𝐵) ≼ ((𝐴 ∖ 𝐵) ⊔ 𝐵))
19	14, 18	eqbrtrrid 5136	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → (𝐴 ∪ 𝐵) ≼ ((𝐴 ∖ 𝐵) ⊔ 𝐵))
20		domtr 8956	. . . . 5 ⊢ ((𝐴 ≼ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ≼ ((𝐴 ∖ 𝐵) ⊔ 𝐵)) → 𝐴 ≼ ((𝐴 ∖ 𝐵) ⊔ 𝐵))
21	13, 19, 20	syl2anc 585	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → 𝐴 ≼ ((𝐴 ∖ 𝐵) ⊔ 𝐵))
22		simp3 1139	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → 𝐵 ≺ 𝐴)
23		sdomdom 8929	. . . . . . . . 9 ⊢ ((𝐴 ∖ 𝐵) ≺ 𝐵 → (𝐴 ∖ 𝐵) ≼ 𝐵)
24		relsdom 8902	. . . . . . . . . 10 ⊢ Rel ≺
25	24	brrelex2i 5689	. . . . . . . . 9 ⊢ ((𝐴 ∖ 𝐵) ≺ 𝐵 → 𝐵 ∈ V)
26		djudom1 10105	. . . . . . . . 9 ⊢ (((𝐴 ∖ 𝐵) ≼ 𝐵 ∧ 𝐵 ∈ V) → ((𝐴 ∖ 𝐵) ⊔ 𝐵) ≼ (𝐵 ⊔ 𝐵))
27	23, 25, 26	syl2anc 585	. . . . . . . 8 ⊢ ((𝐴 ∖ 𝐵) ≺ 𝐵 → ((𝐴 ∖ 𝐵) ⊔ 𝐵) ≼ (𝐵 ⊔ 𝐵))
28		domtr 8956	. . . . . . . . . . 11 ⊢ ((𝐴 ≼ ((𝐴 ∖ 𝐵) ⊔ 𝐵) ∧ ((𝐴 ∖ 𝐵) ⊔ 𝐵) ≼ (𝐵 ⊔ 𝐵)) → 𝐴 ≼ (𝐵 ⊔ 𝐵))
29	28	ex 412	. . . . . . . . . 10 ⊢ (𝐴 ≼ ((𝐴 ∖ 𝐵) ⊔ 𝐵) → (((𝐴 ∖ 𝐵) ⊔ 𝐵) ≼ (𝐵 ⊔ 𝐵) → 𝐴 ≼ (𝐵 ⊔ 𝐵)))
30	21, 29	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → (((𝐴 ∖ 𝐵) ⊔ 𝐵) ≼ (𝐵 ⊔ 𝐵) → 𝐴 ≼ (𝐵 ⊔ 𝐵)))
31		simp2 1138	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → ω ≼ 𝐴)
32		domtr 8956	. . . . . . . . . . . . 13 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ≼ (𝐵 ⊔ 𝐵)) → ω ≼ (𝐵 ⊔ 𝐵))
33	32	ex 412	. . . . . . . . . . . 12 ⊢ (ω ≼ 𝐴 → (𝐴 ≼ (𝐵 ⊔ 𝐵) → ω ≼ (𝐵 ⊔ 𝐵)))
34	31, 33	syl 17	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → (𝐴 ≼ (𝐵 ⊔ 𝐵) → ω ≼ (𝐵 ⊔ 𝐵)))
35		djuinf 10111	. . . . . . . . . . . . 13 ⊢ (ω ≼ 𝐵 ↔ ω ≼ (𝐵 ⊔ 𝐵))
36	35	biimpri 228	. . . . . . . . . . . 12 ⊢ (ω ≼ (𝐵 ⊔ 𝐵) → ω ≼ 𝐵)
37		domrefg 8936	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ dom card → 𝐵 ≼ 𝐵)
38		infdjuabs 10127	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ dom card ∧ ω ≼ 𝐵 ∧ 𝐵 ≼ 𝐵) → (𝐵 ⊔ 𝐵) ≈ 𝐵)
39	38	3com23 1127	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ dom card ∧ 𝐵 ≼ 𝐵 ∧ ω ≼ 𝐵) → (𝐵 ⊔ 𝐵) ≈ 𝐵)
40	39	3expia 1122	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ dom card ∧ 𝐵 ≼ 𝐵) → (ω ≼ 𝐵 → (𝐵 ⊔ 𝐵) ≈ 𝐵))
41	37, 40	mpdan 688	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ dom card → (ω ≼ 𝐵 → (𝐵 ⊔ 𝐵) ≈ 𝐵))
42	8, 36, 41	syl2im 40	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → (ω ≼ (𝐵 ⊔ 𝐵) → (𝐵 ⊔ 𝐵) ≈ 𝐵))
43	34, 42	syld 47	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → (𝐴 ≼ (𝐵 ⊔ 𝐵) → (𝐵 ⊔ 𝐵) ≈ 𝐵))
44		domen2 9060	. . . . . . . . . . 11 ⊢ ((𝐵 ⊔ 𝐵) ≈ 𝐵 → (𝐴 ≼ (𝐵 ⊔ 𝐵) ↔ 𝐴 ≼ 𝐵))
45	44	biimpcd 249	. . . . . . . . . 10 ⊢ (𝐴 ≼ (𝐵 ⊔ 𝐵) → ((𝐵 ⊔ 𝐵) ≈ 𝐵 → 𝐴 ≼ 𝐵))
46	43, 45	sylcom 30	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → (𝐴 ≼ (𝐵 ⊔ 𝐵) → 𝐴 ≼ 𝐵))
47	30, 46	syld 47	. . . . . . . 8 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → (((𝐴 ∖ 𝐵) ⊔ 𝐵) ≼ (𝐵 ⊔ 𝐵) → 𝐴 ≼ 𝐵))
48		domnsym 9043	. . . . . . . 8 ⊢ (𝐴 ≼ 𝐵 → ¬ 𝐵 ≺ 𝐴)
49	27, 47, 48	syl56 36	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → ((𝐴 ∖ 𝐵) ≺ 𝐵 → ¬ 𝐵 ≺ 𝐴))
50	22, 49	mt2d 136	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → ¬ (𝐴 ∖ 𝐵) ≺ 𝐵)
51		domtri2 9913	. . . . . . 7 ⊢ ((𝐵 ∈ dom card ∧ (𝐴 ∖ 𝐵) ∈ dom card) → (𝐵 ≼ (𝐴 ∖ 𝐵) ↔ ¬ (𝐴 ∖ 𝐵) ≺ 𝐵))
52	8, 16, 51	syl2anc 585	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → (𝐵 ≼ (𝐴 ∖ 𝐵) ↔ ¬ (𝐴 ∖ 𝐵) ≺ 𝐵))
53	50, 52	mpbird 257	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → 𝐵 ≼ (𝐴 ∖ 𝐵))
54	1	difexd 5278	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → (𝐴 ∖ 𝐵) ∈ V)
55		djudom2 10106	. . . . 5 ⊢ ((𝐵 ≼ (𝐴 ∖ 𝐵) ∧ (𝐴 ∖ 𝐵) ∈ V) → ((𝐴 ∖ 𝐵) ⊔ 𝐵) ≼ ((𝐴 ∖ 𝐵) ⊔ (𝐴 ∖ 𝐵)))
56	53, 54, 55	syl2anc 585	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → ((𝐴 ∖ 𝐵) ⊔ 𝐵) ≼ ((𝐴 ∖ 𝐵) ⊔ (𝐴 ∖ 𝐵)))
57		domtr 8956	. . . 4 ⊢ ((𝐴 ≼ ((𝐴 ∖ 𝐵) ⊔ 𝐵) ∧ ((𝐴 ∖ 𝐵) ⊔ 𝐵) ≼ ((𝐴 ∖ 𝐵) ⊔ (𝐴 ∖ 𝐵))) → 𝐴 ≼ ((𝐴 ∖ 𝐵) ⊔ (𝐴 ∖ 𝐵)))
58	21, 56, 57	syl2anc 585	. . 3 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → 𝐴 ≼ ((𝐴 ∖ 𝐵) ⊔ (𝐴 ∖ 𝐵)))
59		domtr 8956	. . . . . 6 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ≼ ((𝐴 ∖ 𝐵) ⊔ (𝐴 ∖ 𝐵))) → ω ≼ ((𝐴 ∖ 𝐵) ⊔ (𝐴 ∖ 𝐵)))
60	31, 58, 59	syl2anc 585	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → ω ≼ ((𝐴 ∖ 𝐵) ⊔ (𝐴 ∖ 𝐵)))
61		djuinf 10111	. . . . 5 ⊢ (ω ≼ (𝐴 ∖ 𝐵) ↔ ω ≼ ((𝐴 ∖ 𝐵) ⊔ (𝐴 ∖ 𝐵)))
62	60, 61	sylibr 234	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → ω ≼ (𝐴 ∖ 𝐵))
63		domrefg 8936	. . . . 5 ⊢ ((𝐴 ∖ 𝐵) ∈ dom card → (𝐴 ∖ 𝐵) ≼ (𝐴 ∖ 𝐵))
64	16, 63	syl 17	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → (𝐴 ∖ 𝐵) ≼ (𝐴 ∖ 𝐵))
65		infdjuabs 10127	. . . 4 ⊢ (((𝐴 ∖ 𝐵) ∈ dom card ∧ ω ≼ (𝐴 ∖ 𝐵) ∧ (𝐴 ∖ 𝐵) ≼ (𝐴 ∖ 𝐵)) → ((𝐴 ∖ 𝐵) ⊔ (𝐴 ∖ 𝐵)) ≈ (𝐴 ∖ 𝐵))
66	16, 62, 64, 65	syl3anc 1374	. . 3 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → ((𝐴 ∖ 𝐵) ⊔ (𝐴 ∖ 𝐵)) ≈ (𝐴 ∖ 𝐵))
67		domentr 8962	. . 3 ⊢ ((𝐴 ≼ ((𝐴 ∖ 𝐵) ⊔ (𝐴 ∖ 𝐵)) ∧ ((𝐴 ∖ 𝐵) ⊔ (𝐴 ∖ 𝐵)) ≈ (𝐴 ∖ 𝐵)) → 𝐴 ≼ (𝐴 ∖ 𝐵))
68	58, 66, 67	syl2anc 585	. 2 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → 𝐴 ≼ (𝐴 ∖ 𝐵))
69		sbth 9037	. 2 ⊢ (((𝐴 ∖ 𝐵) ≼ 𝐴 ∧ 𝐴 ≼ (𝐴 ∖ 𝐵)) → (𝐴 ∖ 𝐵) ≈ 𝐴)
70	4, 68, 69	syl2anc 585	1 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → (𝐴 ∖ 𝐵) ≈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ w3a 1087   ∈ wcel 2114  Vcvv 3442   ∖ cdif 3900   ∪ cun 3901   ⊆ wss 3903   class class class wbr 5100  dom cdm 5632  ωcom 7818   ≈ cen 8892   ≼ cdom 8893   ≺ csdm 8894   ⊔ cdju 9822  cardccrd 9859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-inf2 9562

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-oadd 8411  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-oi 9427  df-dju 9825  df-card 9863

This theorem is referenced by:  infdif2  10131  alephsuc3  10503  aleph1irr  16183