Theorem infdif 10277

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 1136	. . 3 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → 𝐴 ∈ dom card)
2		difss 4159	. . 3 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
3		ssdomg 9060	. . 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 9040	. . . . . . . . 9 ⊢ (𝐵 ≺ 𝐴 → 𝐵 ≼ 𝐴)
6	5	3ad2ant3 1135	. . . . . . . 8 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → 𝐵 ≼ 𝐴)
7		numdom 10107	. . . . . . . 8 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ dom card)
8	1, 6, 7	syl2anc 583	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → 𝐵 ∈ dom card)
9		unnum 10266	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ∪ 𝐵) ∈ dom card)
10	1, 8, 9	syl2anc 583	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → (𝐴 ∪ 𝐵) ∈ dom card)
11		ssun1 4201	. . . . . 6 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
12		ssdomg 9060	. . . . . 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 4499	. . . . . 6 ⊢ ((𝐴 ∖ 𝐵) ∪ 𝐵) = (𝐴 ∪ 𝐵)
15		ssnum 10108	. . . . . . . 8 ⊢ ((𝐴 ∈ dom card ∧ (𝐴 ∖ 𝐵) ⊆ 𝐴) → (𝐴 ∖ 𝐵) ∈ dom card)
16	1, 2, 15	sylancl 585	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → (𝐴 ∖ 𝐵) ∈ dom card)
17		undjudom 10237	. . . . . . 7 ⊢ (((𝐴 ∖ 𝐵) ∈ dom card ∧ 𝐵 ∈ dom card) → ((𝐴 ∖ 𝐵) ∪ 𝐵) ≼ ((𝐴 ∖ 𝐵) ⊔ 𝐵))
18	16, 8, 17	syl2anc 583	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → ((𝐴 ∖ 𝐵) ∪ 𝐵) ≼ ((𝐴 ∖ 𝐵) ⊔ 𝐵))
19	14, 18	eqbrtrrid 5202	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → (𝐴 ∪ 𝐵) ≼ ((𝐴 ∖ 𝐵) ⊔ 𝐵))
20		domtr 9067	. . . . 5 ⊢ ((𝐴 ≼ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ≼ ((𝐴 ∖ 𝐵) ⊔ 𝐵)) → 𝐴 ≼ ((𝐴 ∖ 𝐵) ⊔ 𝐵))
21	13, 19, 20	syl2anc 583	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → 𝐴 ≼ ((𝐴 ∖ 𝐵) ⊔ 𝐵))
22		simp3 1138	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → 𝐵 ≺ 𝐴)
23		sdomdom 9040	. . . . . . . . 9 ⊢ ((𝐴 ∖ 𝐵) ≺ 𝐵 → (𝐴 ∖ 𝐵) ≼ 𝐵)
24		relsdom 9010	. . . . . . . . . 10 ⊢ Rel ≺
25	24	brrelex2i 5757	. . . . . . . . 9 ⊢ ((𝐴 ∖ 𝐵) ≺ 𝐵 → 𝐵 ∈ V)
26		djudom1 10252	. . . . . . . . 9 ⊢ (((𝐴 ∖ 𝐵) ≼ 𝐵 ∧ 𝐵 ∈ V) → ((𝐴 ∖ 𝐵) ⊔ 𝐵) ≼ (𝐵 ⊔ 𝐵))
27	23, 25, 26	syl2anc 583	. . . . . . . 8 ⊢ ((𝐴 ∖ 𝐵) ≺ 𝐵 → ((𝐴 ∖ 𝐵) ⊔ 𝐵) ≼ (𝐵 ⊔ 𝐵))
28		domtr 9067	. . . . . . . . . . 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 1137	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → ω ≼ 𝐴)
32		domtr 9067	. . . . . . . . . . . . 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 10258	. . . . . . . . . . . . 13 ⊢ (ω ≼ 𝐵 ↔ ω ≼ (𝐵 ⊔ 𝐵))
36	35	biimpri 228	. . . . . . . . . . . 12 ⊢ (ω ≼ (𝐵 ⊔ 𝐵) → ω ≼ 𝐵)
37		domrefg 9047	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ dom card → 𝐵 ≼ 𝐵)
38		infdjuabs 10274	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ dom card ∧ ω ≼ 𝐵 ∧ 𝐵 ≼ 𝐵) → (𝐵 ⊔ 𝐵) ≈ 𝐵)
39	38	3com23 1126	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ dom card ∧ 𝐵 ≼ 𝐵 ∧ ω ≼ 𝐵) → (𝐵 ⊔ 𝐵) ≈ 𝐵)
40	39	3expia 1121	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ dom card ∧ 𝐵 ≼ 𝐵) → (ω ≼ 𝐵 → (𝐵 ⊔ 𝐵) ≈ 𝐵))
41	37, 40	mpdan 686	. . . . . . . . . . . 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 9186	. . . . . . . . . . 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 9165	. . . . . . . 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 10058	. . . . . . 7 ⊢ ((𝐵 ∈ dom card ∧ (𝐴 ∖ 𝐵) ∈ dom card) → (𝐵 ≼ (𝐴 ∖ 𝐵) ↔ ¬ (𝐴 ∖ 𝐵) ≺ 𝐵))
52	8, 16, 51	syl2anc 583	. . . . . 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 5349	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → (𝐴 ∖ 𝐵) ∈ V)
55		djudom2 10253	. . . . 5 ⊢ ((𝐵 ≼ (𝐴 ∖ 𝐵) ∧ (𝐴 ∖ 𝐵) ∈ V) → ((𝐴 ∖ 𝐵) ⊔ 𝐵) ≼ ((𝐴 ∖ 𝐵) ⊔ (𝐴 ∖ 𝐵)))
56	53, 54, 55	syl2anc 583	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → ((𝐴 ∖ 𝐵) ⊔ 𝐵) ≼ ((𝐴 ∖ 𝐵) ⊔ (𝐴 ∖ 𝐵)))
57		domtr 9067	. . . 4 ⊢ ((𝐴 ≼ ((𝐴 ∖ 𝐵) ⊔ 𝐵) ∧ ((𝐴 ∖ 𝐵) ⊔ 𝐵) ≼ ((𝐴 ∖ 𝐵) ⊔ (𝐴 ∖ 𝐵))) → 𝐴 ≼ ((𝐴 ∖ 𝐵) ⊔ (𝐴 ∖ 𝐵)))
58	21, 56, 57	syl2anc 583	. . 3 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → 𝐴 ≼ ((𝐴 ∖ 𝐵) ⊔ (𝐴 ∖ 𝐵)))
59		domtr 9067	. . . . . 6 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ≼ ((𝐴 ∖ 𝐵) ⊔ (𝐴 ∖ 𝐵))) → ω ≼ ((𝐴 ∖ 𝐵) ⊔ (𝐴 ∖ 𝐵)))
60	31, 58, 59	syl2anc 583	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → ω ≼ ((𝐴 ∖ 𝐵) ⊔ (𝐴 ∖ 𝐵)))
61		djuinf 10258	. . . . 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 9047	. . . . 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 10274	. . . 4 ⊢ (((𝐴 ∖ 𝐵) ∈ dom card ∧ ω ≼ (𝐴 ∖ 𝐵) ∧ (𝐴 ∖ 𝐵) ≼ (𝐴 ∖ 𝐵)) → ((𝐴 ∖ 𝐵) ⊔ (𝐴 ∖ 𝐵)) ≈ (𝐴 ∖ 𝐵))
66	16, 62, 64, 65	syl3anc 1371	. . 3 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → ((𝐴 ∖ 𝐵) ⊔ (𝐴 ∖ 𝐵)) ≈ (𝐴 ∖ 𝐵))
67		domentr 9073	. . 3 ⊢ ((𝐴 ≼ ((𝐴 ∖ 𝐵) ⊔ (𝐴 ∖ 𝐵)) ∧ ((𝐴 ∖ 𝐵) ⊔ (𝐴 ∖ 𝐵)) ≈ (𝐴 ∖ 𝐵)) → 𝐴 ≼ (𝐴 ∖ 𝐵))
68	58, 66, 67	syl2anc 583	. 2 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → 𝐴 ≼ (𝐴 ∖ 𝐵))
69		sbth 9159	. 2 ⊢ (((𝐴 ∖ 𝐵) ≼ 𝐴 ∧ 𝐴 ≼ (𝐴 ∖ 𝐵)) → (𝐴 ∖ 𝐵) ≈ 𝐴)
70	4, 68, 69	syl2anc 583	1 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → (𝐴 ∖ 𝐵) ≈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ w3a 1087   ∈ wcel 2108  Vcvv 3488   ∖ cdif 3973   ∪ cun 3974   ⊆ wss 3976   class class class wbr 5166  dom cdm 5700  ωcom 7903   ≈ cen 9000   ≼ cdom 9001   ≺ csdm 9002   ⊔ cdju 9967  cardccrd 10004

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-oadd 8526  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-oi 9579  df-dju 9970  df-card 10008

This theorem is referenced by:  infdif2  10278  alephsuc3  10649  aleph1irr  16294