Theorem infdif 10122

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 1142	. . 3 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → 𝐴 ∈ dom card)
2		difss 4067	. . 3 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
3		ssdomg 8938	. . 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 8918	. . . . . . . . 9 ⊢ (𝐵 ≺ 𝐴 → 𝐵 ≼ 𝐴)
6	5	3ad2ant3 1141	. . . . . . . 8 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → 𝐵 ≼ 𝐴)
7		numdom 9952	. . . . . . . 8 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ dom card)
8	1, 6, 7	syl2anc 590	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → 𝐵 ∈ dom card)
9		unnum 10111	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ∪ 𝐵) ∈ dom card)
10	1, 8, 9	syl2anc 590	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → (𝐴 ∪ 𝐵) ∈ dom card)
11		ssun1 4108	. . . . . 6 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
12		ssdomg 8938	. . . . . 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 4405	. . . . . 6 ⊢ ((𝐴 ∖ 𝐵) ∪ 𝐵) = (𝐴 ∪ 𝐵)
15		ssnum 9953	. . . . . . . 8 ⊢ ((𝐴 ∈ dom card ∧ (𝐴 ∖ 𝐵) ⊆ 𝐴) → (𝐴 ∖ 𝐵) ∈ dom card)
16	1, 2, 15	sylancl 592	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → (𝐴 ∖ 𝐵) ∈ dom card)
17		undjudom 10082	. . . . . . 7 ⊢ (((𝐴 ∖ 𝐵) ∈ dom card ∧ 𝐵 ∈ dom card) → ((𝐴 ∖ 𝐵) ∪ 𝐵) ≼ ((𝐴 ∖ 𝐵) ⊔ 𝐵))
18	16, 8, 17	syl2anc 590	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → ((𝐴 ∖ 𝐵) ∪ 𝐵) ≼ ((𝐴 ∖ 𝐵) ⊔ 𝐵))
19	14, 18	eqbrtrrid 5109	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → (𝐴 ∪ 𝐵) ≼ ((𝐴 ∖ 𝐵) ⊔ 𝐵))
20		domtr 8945	. . . . 5 ⊢ ((𝐴 ≼ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ≼ ((𝐴 ∖ 𝐵) ⊔ 𝐵)) → 𝐴 ≼ ((𝐴 ∖ 𝐵) ⊔ 𝐵))
21	13, 19, 20	syl2anc 590	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → 𝐴 ≼ ((𝐴 ∖ 𝐵) ⊔ 𝐵))
22		simp3 1144	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → 𝐵 ≺ 𝐴)
23		sdomdom 8918	. . . . . . . . 9 ⊢ ((𝐴 ∖ 𝐵) ≺ 𝐵 → (𝐴 ∖ 𝐵) ≼ 𝐵)
24		relsdom 8891	. . . . . . . . . 10 ⊢ Rel ≺
25	24	brrelex2i 5676	. . . . . . . . 9 ⊢ ((𝐴 ∖ 𝐵) ≺ 𝐵 → 𝐵 ∈ V)
26		djudom1 10097	. . . . . . . . 9 ⊢ (((𝐴 ∖ 𝐵) ≼ 𝐵 ∧ 𝐵 ∈ V) → ((𝐴 ∖ 𝐵) ⊔ 𝐵) ≼ (𝐵 ⊔ 𝐵))
27	23, 25, 26	syl2anc 590	. . . . . . . 8 ⊢ ((𝐴 ∖ 𝐵) ≺ 𝐵 → ((𝐴 ∖ 𝐵) ⊔ 𝐵) ≼ (𝐵 ⊔ 𝐵))
28		domtr 8945	. . . . . . . . . . 11 ⊢ ((𝐴 ≼ ((𝐴 ∖ 𝐵) ⊔ 𝐵) ∧ ((𝐴 ∖ 𝐵) ⊔ 𝐵) ≼ (𝐵 ⊔ 𝐵)) → 𝐴 ≼ (𝐵 ⊔ 𝐵))
29	28	ex 413	. . . . . . . . . 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 1143	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → ω ≼ 𝐴)
32		domtr 8945	. . . . . . . . . . . . 13 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ≼ (𝐵 ⊔ 𝐵)) → ω ≼ (𝐵 ⊔ 𝐵))
33	32	ex 413	. . . . . . . . . . . 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 10103	. . . . . . . . . . . . 13 ⊢ (ω ≼ 𝐵 ↔ ω ≼ (𝐵 ⊔ 𝐵))
36	35	biimpri 229	. . . . . . . . . . . 12 ⊢ (ω ≼ (𝐵 ⊔ 𝐵) → ω ≼ 𝐵)
37		domrefg 8925	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ dom card → 𝐵 ≼ 𝐵)
38		infdjuabs 10119	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ dom card ∧ ω ≼ 𝐵 ∧ 𝐵 ≼ 𝐵) → (𝐵 ⊔ 𝐵) ≈ 𝐵)
39	38	3com23 1132	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ dom card ∧ 𝐵 ≼ 𝐵 ∧ ω ≼ 𝐵) → (𝐵 ⊔ 𝐵) ≈ 𝐵)
40	39	3expia 1127	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ dom card ∧ 𝐵 ≼ 𝐵) → (ω ≼ 𝐵 → (𝐵 ⊔ 𝐵) ≈ 𝐵))
41	37, 40	mpdan 693	. . . . . . . . . . . 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 9049	. . . . . . . . . . 11 ⊢ ((𝐵 ⊔ 𝐵) ≈ 𝐵 → (𝐴 ≼ (𝐵 ⊔ 𝐵) ↔ 𝐴 ≼ 𝐵))
45	44	biimpcd 250	. . . . . . . . . 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 9032	. . . . . . . 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 9905	. . . . . . 7 ⊢ ((𝐵 ∈ dom card ∧ (𝐴 ∖ 𝐵) ∈ dom card) → (𝐵 ≼ (𝐴 ∖ 𝐵) ↔ ¬ (𝐴 ∖ 𝐵) ≺ 𝐵))
52	8, 16, 51	syl2anc 590	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → (𝐵 ≼ (𝐴 ∖ 𝐵) ↔ ¬ (𝐴 ∖ 𝐵) ≺ 𝐵))
53	50, 52	mpbird 258	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → 𝐵 ≼ (𝐴 ∖ 𝐵))
54	1	difexd 5260	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → (𝐴 ∖ 𝐵) ∈ V)
55		djudom2 10098	. . . . 5 ⊢ ((𝐵 ≼ (𝐴 ∖ 𝐵) ∧ (𝐴 ∖ 𝐵) ∈ V) → ((𝐴 ∖ 𝐵) ⊔ 𝐵) ≼ ((𝐴 ∖ 𝐵) ⊔ (𝐴 ∖ 𝐵)))
56	53, 54, 55	syl2anc 590	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → ((𝐴 ∖ 𝐵) ⊔ 𝐵) ≼ ((𝐴 ∖ 𝐵) ⊔ (𝐴 ∖ 𝐵)))
57		domtr 8945	. . . 4 ⊢ ((𝐴 ≼ ((𝐴 ∖ 𝐵) ⊔ 𝐵) ∧ ((𝐴 ∖ 𝐵) ⊔ 𝐵) ≼ ((𝐴 ∖ 𝐵) ⊔ (𝐴 ∖ 𝐵))) → 𝐴 ≼ ((𝐴 ∖ 𝐵) ⊔ (𝐴 ∖ 𝐵)))
58	21, 56, 57	syl2anc 590	. . 3 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → 𝐴 ≼ ((𝐴 ∖ 𝐵) ⊔ (𝐴 ∖ 𝐵)))
59		domtr 8945	. . . . . 6 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ≼ ((𝐴 ∖ 𝐵) ⊔ (𝐴 ∖ 𝐵))) → ω ≼ ((𝐴 ∖ 𝐵) ⊔ (𝐴 ∖ 𝐵)))
60	31, 58, 59	syl2anc 590	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → ω ≼ ((𝐴 ∖ 𝐵) ⊔ (𝐴 ∖ 𝐵)))
61		djuinf 10103	. . . . 5 ⊢ (ω ≼ (𝐴 ∖ 𝐵) ↔ ω ≼ ((𝐴 ∖ 𝐵) ⊔ (𝐴 ∖ 𝐵)))
62	60, 61	sylibr 235	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → ω ≼ (𝐴 ∖ 𝐵))
63		domrefg 8925	. . . . 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 10119	. . . 4 ⊢ (((𝐴 ∖ 𝐵) ∈ dom card ∧ ω ≼ (𝐴 ∖ 𝐵) ∧ (𝐴 ∖ 𝐵) ≼ (𝐴 ∖ 𝐵)) → ((𝐴 ∖ 𝐵) ⊔ (𝐴 ∖ 𝐵)) ≈ (𝐴 ∖ 𝐵))
66	16, 62, 64, 65	syl3anc 1379	. . 3 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → ((𝐴 ∖ 𝐵) ⊔ (𝐴 ∖ 𝐵)) ≈ (𝐴 ∖ 𝐵))
67		domentr 8951	. . 3 ⊢ ((𝐴 ≼ ((𝐴 ∖ 𝐵) ⊔ (𝐴 ∖ 𝐵)) ∧ ((𝐴 ∖ 𝐵) ⊔ (𝐴 ∖ 𝐵)) ≈ (𝐴 ∖ 𝐵)) → 𝐴 ≼ (𝐴 ∖ 𝐵))
68	58, 66, 67	syl2anc 590	. 2 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → 𝐴 ≼ (𝐴 ∖ 𝐵))
69		sbth 9026	. 2 ⊢ (((𝐴 ∖ 𝐵) ≼ 𝐴 ∧ 𝐴 ≼ (𝐴 ∖ 𝐵)) → (𝐴 ∖ 𝐵) ≈ 𝐴)
70	4, 68, 69	syl2anc 590	1 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → (𝐴 ∖ 𝐵) ≈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ w3a 1092   ∈ wcel 2119  Vcvv 3431   ∖ cdif 3880   ∪ cun 3881   ⊆ wss 3883   class class class wbr 5073  dom cdm 5619  ωcom 7807   ≈ cen 8881   ≼ cdom 8882   ≺ csdm 8883   ⊔ cdju 9814  cardccrd 9851

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5200  ax-sep 5219  ax-nul 5229  ax-pow 5295  ax-pr 5363  ax-un 7679  ax-inf2 9554

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-int 4879  df-iun 4924  df-br 5074  df-opab 5136  df-mpt 5155  df-tr 5181  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7314  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7808  df-1st 7932  df-2nd 7933  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-1o 8396  df-2o 8397  df-oadd 8400  df-er 8634  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-oi 9416  df-dju 9817  df-card 9855

This theorem is referenced by:  infdif2  10123  alephsuc3  10495  aleph1irr  16205