Theorem infdif 8914

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 1054	. . 3 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → 𝐴 ∈ dom card)
2		difss 3699	. . 3 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
3		ssdomg 7887	. . 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 7869	. . . . . . . . 9 ⊢ (𝐵 ≺ 𝐴 → 𝐵 ≼ 𝐴)
6	5	3ad2ant3 1077	. . . . . . . 8 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → 𝐵 ≼ 𝐴)
7		numdom 8744	. . . . . . . 8 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ dom card)
8	1, 6, 7	syl2anc 691	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → 𝐵 ∈ dom card)
9		unnum 8905	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ∪ 𝐵) ∈ dom card)
10	1, 8, 9	syl2anc 691	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → (𝐴 ∪ 𝐵) ∈ dom card)
11		ssun1 3738	. . . . . 6 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
12		ssdomg 7887	. . . . . 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 3995	. . . . . 6 ⊢ ((𝐴 ∖ 𝐵) ∪ 𝐵) = (𝐴 ∪ 𝐵)
15		ssnum 8745	. . . . . . . 8 ⊢ ((𝐴 ∈ dom card ∧ (𝐴 ∖ 𝐵) ⊆ 𝐴) → (𝐴 ∖ 𝐵) ∈ dom card)
16	1, 2, 15	sylancl 693	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → (𝐴 ∖ 𝐵) ∈ dom card)
17		uncdadom 8876	. . . . . . 7 ⊢ (((𝐴 ∖ 𝐵) ∈ dom card ∧ 𝐵 ∈ dom card) → ((𝐴 ∖ 𝐵) ∪ 𝐵) ≼ ((𝐴 ∖ 𝐵) +𝑐 𝐵))
18	16, 8, 17	syl2anc 691	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → ((𝐴 ∖ 𝐵) ∪ 𝐵) ≼ ((𝐴 ∖ 𝐵) +𝑐 𝐵))
19	14, 18	syl5eqbrr 4619	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → (𝐴 ∪ 𝐵) ≼ ((𝐴 ∖ 𝐵) +𝑐 𝐵))
20		domtr 7895	. . . . 5 ⊢ ((𝐴 ≼ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ≼ ((𝐴 ∖ 𝐵) +𝑐 𝐵)) → 𝐴 ≼ ((𝐴 ∖ 𝐵) +𝑐 𝐵))
21	13, 19, 20	syl2anc 691	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → 𝐴 ≼ ((𝐴 ∖ 𝐵) +𝑐 𝐵))
22		simp3 1056	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → 𝐵 ≺ 𝐴)
23		sdomdom 7869	. . . . . . . . 9 ⊢ ((𝐴 ∖ 𝐵) ≺ 𝐵 → (𝐴 ∖ 𝐵) ≼ 𝐵)
24		cdadom1 8891	. . . . . . . . 9 ⊢ ((𝐴 ∖ 𝐵) ≼ 𝐵 → ((𝐴 ∖ 𝐵) +𝑐 𝐵) ≼ (𝐵 +𝑐 𝐵))
25	23, 24	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∖ 𝐵) ≺ 𝐵 → ((𝐴 ∖ 𝐵) +𝑐 𝐵) ≼ (𝐵 +𝑐 𝐵))
26		domtr 7895	. . . . . . . . . . 11 ⊢ ((𝐴 ≼ ((𝐴 ∖ 𝐵) +𝑐 𝐵) ∧ ((𝐴 ∖ 𝐵) +𝑐 𝐵) ≼ (𝐵 +𝑐 𝐵)) → 𝐴 ≼ (𝐵 +𝑐 𝐵))
27	26	ex 449	. . . . . . . . . 10 ⊢ (𝐴 ≼ ((𝐴 ∖ 𝐵) +𝑐 𝐵) → (((𝐴 ∖ 𝐵) +𝑐 𝐵) ≼ (𝐵 +𝑐 𝐵) → 𝐴 ≼ (𝐵 +𝑐 𝐵)))
28	21, 27	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → (((𝐴 ∖ 𝐵) +𝑐 𝐵) ≼ (𝐵 +𝑐 𝐵) → 𝐴 ≼ (𝐵 +𝑐 𝐵)))
29		simp2 1055	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → ω ≼ 𝐴)
30		domtr 7895	. . . . . . . . . . . . 13 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ≼ (𝐵 +𝑐 𝐵)) → ω ≼ (𝐵 +𝑐 𝐵))
31	30	ex 449	. . . . . . . . . . . 12 ⊢ (ω ≼ 𝐴 → (𝐴 ≼ (𝐵 +𝑐 𝐵) → ω ≼ (𝐵 +𝑐 𝐵)))
32	29, 31	syl 17	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → (𝐴 ≼ (𝐵 +𝑐 𝐵) → ω ≼ (𝐵 +𝑐 𝐵)))
33		cdainf 8897	. . . . . . . . . . . . 13 ⊢ (ω ≼ 𝐵 ↔ ω ≼ (𝐵 +𝑐 𝐵))
34	33	biimpri 217	. . . . . . . . . . . 12 ⊢ (ω ≼ (𝐵 +𝑐 𝐵) → ω ≼ 𝐵)
35		domrefg 7876	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ dom card → 𝐵 ≼ 𝐵)
36		infcdaabs 8911	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ dom card ∧ ω ≼ 𝐵 ∧ 𝐵 ≼ 𝐵) → (𝐵 +𝑐 𝐵) ≈ 𝐵)
37	36	3com23 1263	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ dom card ∧ 𝐵 ≼ 𝐵 ∧ ω ≼ 𝐵) → (𝐵 +𝑐 𝐵) ≈ 𝐵)
38	37	3expia 1259	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ dom card ∧ 𝐵 ≼ 𝐵) → (ω ≼ 𝐵 → (𝐵 +𝑐 𝐵) ≈ 𝐵))
39	35, 38	mpdan 699	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ dom card → (ω ≼ 𝐵 → (𝐵 +𝑐 𝐵) ≈ 𝐵))
40	8, 34, 39	syl2im 39	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → (ω ≼ (𝐵 +𝑐 𝐵) → (𝐵 +𝑐 𝐵) ≈ 𝐵))
41	32, 40	syld 46	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → (𝐴 ≼ (𝐵 +𝑐 𝐵) → (𝐵 +𝑐 𝐵) ≈ 𝐵))
42		domen2 7988	. . . . . . . . . . 11 ⊢ ((𝐵 +𝑐 𝐵) ≈ 𝐵 → (𝐴 ≼ (𝐵 +𝑐 𝐵) ↔ 𝐴 ≼ 𝐵))
43	42	biimpcd 238	. . . . . . . . . 10 ⊢ (𝐴 ≼ (𝐵 +𝑐 𝐵) → ((𝐵 +𝑐 𝐵) ≈ 𝐵 → 𝐴 ≼ 𝐵))
44	41, 43	sylcom 30	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → (𝐴 ≼ (𝐵 +𝑐 𝐵) → 𝐴 ≼ 𝐵))
45	28, 44	syld 46	. . . . . . . 8 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → (((𝐴 ∖ 𝐵) +𝑐 𝐵) ≼ (𝐵 +𝑐 𝐵) → 𝐴 ≼ 𝐵))
46		domnsym 7971	. . . . . . . 8 ⊢ (𝐴 ≼ 𝐵 → ¬ 𝐵 ≺ 𝐴)
47	25, 45, 46	syl56 35	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → ((𝐴 ∖ 𝐵) ≺ 𝐵 → ¬ 𝐵 ≺ 𝐴))
48	22, 47	mt2d 130	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → ¬ (𝐴 ∖ 𝐵) ≺ 𝐵)
49		domtri2 8698	. . . . . . 7 ⊢ ((𝐵 ∈ dom card ∧ (𝐴 ∖ 𝐵) ∈ dom card) → (𝐵 ≼ (𝐴 ∖ 𝐵) ↔ ¬ (𝐴 ∖ 𝐵) ≺ 𝐵))
50	8, 16, 49	syl2anc 691	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → (𝐵 ≼ (𝐴 ∖ 𝐵) ↔ ¬ (𝐴 ∖ 𝐵) ≺ 𝐵))
51	48, 50	mpbird 246	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → 𝐵 ≼ (𝐴 ∖ 𝐵))
52		cdadom2 8892	. . . . 5 ⊢ (𝐵 ≼ (𝐴 ∖ 𝐵) → ((𝐴 ∖ 𝐵) +𝑐 𝐵) ≼ ((𝐴 ∖ 𝐵) +𝑐 (𝐴 ∖ 𝐵)))
53	51, 52	syl 17	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → ((𝐴 ∖ 𝐵) +𝑐 𝐵) ≼ ((𝐴 ∖ 𝐵) +𝑐 (𝐴 ∖ 𝐵)))
54		domtr 7895	. . . 4 ⊢ ((𝐴 ≼ ((𝐴 ∖ 𝐵) +𝑐 𝐵) ∧ ((𝐴 ∖ 𝐵) +𝑐 𝐵) ≼ ((𝐴 ∖ 𝐵) +𝑐 (𝐴 ∖ 𝐵))) → 𝐴 ≼ ((𝐴 ∖ 𝐵) +𝑐 (𝐴 ∖ 𝐵)))
55	21, 53, 54	syl2anc 691	. . 3 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → 𝐴 ≼ ((𝐴 ∖ 𝐵) +𝑐 (𝐴 ∖ 𝐵)))
56		domtr 7895	. . . . . 6 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ≼ ((𝐴 ∖ 𝐵) +𝑐 (𝐴 ∖ 𝐵))) → ω ≼ ((𝐴 ∖ 𝐵) +𝑐 (𝐴 ∖ 𝐵)))
57	29, 55, 56	syl2anc 691	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → ω ≼ ((𝐴 ∖ 𝐵) +𝑐 (𝐴 ∖ 𝐵)))
58		cdainf 8897	. . . . 5 ⊢ (ω ≼ (𝐴 ∖ 𝐵) ↔ ω ≼ ((𝐴 ∖ 𝐵) +𝑐 (𝐴 ∖ 𝐵)))
59	57, 58	sylibr 223	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → ω ≼ (𝐴 ∖ 𝐵))
60		domrefg 7876	. . . . 5 ⊢ ((𝐴 ∖ 𝐵) ∈ dom card → (𝐴 ∖ 𝐵) ≼ (𝐴 ∖ 𝐵))
61	16, 60	syl 17	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → (𝐴 ∖ 𝐵) ≼ (𝐴 ∖ 𝐵))
62		infcdaabs 8911	. . . 4 ⊢ (((𝐴 ∖ 𝐵) ∈ dom card ∧ ω ≼ (𝐴 ∖ 𝐵) ∧ (𝐴 ∖ 𝐵) ≼ (𝐴 ∖ 𝐵)) → ((𝐴 ∖ 𝐵) +𝑐 (𝐴 ∖ 𝐵)) ≈ (𝐴 ∖ 𝐵))
63	16, 59, 61, 62	syl3anc 1318	. . 3 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → ((𝐴 ∖ 𝐵) +𝑐 (𝐴 ∖ 𝐵)) ≈ (𝐴 ∖ 𝐵))
64		domentr 7901	. . 3 ⊢ ((𝐴 ≼ ((𝐴 ∖ 𝐵) +𝑐 (𝐴 ∖ 𝐵)) ∧ ((𝐴 ∖ 𝐵) +𝑐 (𝐴 ∖ 𝐵)) ≈ (𝐴 ∖ 𝐵)) → 𝐴 ≼ (𝐴 ∖ 𝐵))
65	55, 63, 64	syl2anc 691	. 2 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → 𝐴 ≼ (𝐴 ∖ 𝐵))
66		sbth 7965	. 2 ⊢ (((𝐴 ∖ 𝐵) ≼ 𝐴 ∧ 𝐴 ≼ (𝐴 ∖ 𝐵)) → (𝐴 ∖ 𝐵) ≈ 𝐴)
67	4, 65, 66	syl2anc 691	1 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → (𝐴 ∖ 𝐵) ≈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ w3a 1031   ∈ wcel 1977   ∖ cdif 3537   ∪ cun 3538   ⊆ wss 3540   class class class wbr 4583  dom cdm 5038  (class class class)co 6549  ωcom 6957   ≈ cen 7838   ≼ cdom 7839   ≺ csdm 7840  cardccrd 8644   +_𝑐 ccda 8872

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-oi 8298  df-card 8648  df-cda 8873

This theorem is referenced by:  infdif2  8915  alephsuc3  9281  aleph1irr  14814