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Theorem infdif 8886
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 ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≈ 𝐴)

Proof of Theorem infdif
StepHypRef Expression
1 simp1 1053 . . 3 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐴 ∈ dom card)
2 difss 3693 . . 3 (𝐴𝐵) ⊆ 𝐴
3 ssdomg 7859 . . 3 (𝐴 ∈ dom card → ((𝐴𝐵) ⊆ 𝐴 → (𝐴𝐵) ≼ 𝐴))
41, 2, 3mpisyl 21 . 2 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≼ 𝐴)
5 sdomdom 7841 . . . . . . . . 9 (𝐵𝐴𝐵𝐴)
653ad2ant3 1076 . . . . . . . 8 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐵𝐴)
7 numdom 8716 . . . . . . . 8 ((𝐴 ∈ dom card ∧ 𝐵𝐴) → 𝐵 ∈ dom card)
81, 6, 7syl2anc 690 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐵 ∈ dom card)
9 unnum 8877 . . . . . . 7 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵) ∈ dom card)
101, 8, 9syl2anc 690 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ∈ dom card)
11 ssun1 3732 . . . . . 6 𝐴 ⊆ (𝐴𝐵)
12 ssdomg 7859 . . . . . 6 ((𝐴𝐵) ∈ dom card → (𝐴 ⊆ (𝐴𝐵) → 𝐴 ≼ (𝐴𝐵)))
1310, 11, 12mpisyl 21 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐴 ≼ (𝐴𝐵))
14 undif1 3989 . . . . . 6 ((𝐴𝐵) ∪ 𝐵) = (𝐴𝐵)
15 ssnum 8717 . . . . . . . 8 ((𝐴 ∈ dom card ∧ (𝐴𝐵) ⊆ 𝐴) → (𝐴𝐵) ∈ dom card)
161, 2, 15sylancl 692 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ∈ dom card)
17 uncdadom 8848 . . . . . . 7 (((𝐴𝐵) ∈ dom card ∧ 𝐵 ∈ dom card) → ((𝐴𝐵) ∪ 𝐵) ≼ ((𝐴𝐵) +𝑐 𝐵))
1816, 8, 17syl2anc 690 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ((𝐴𝐵) ∪ 𝐵) ≼ ((𝐴𝐵) +𝑐 𝐵))
1914, 18syl5eqbrr 4608 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≼ ((𝐴𝐵) +𝑐 𝐵))
20 domtr 7867 . . . . 5 ((𝐴 ≼ (𝐴𝐵) ∧ (𝐴𝐵) ≼ ((𝐴𝐵) +𝑐 𝐵)) → 𝐴 ≼ ((𝐴𝐵) +𝑐 𝐵))
2113, 19, 20syl2anc 690 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐴 ≼ ((𝐴𝐵) +𝑐 𝐵))
22 simp3 1055 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐵𝐴)
23 sdomdom 7841 . . . . . . . . 9 ((𝐴𝐵) ≺ 𝐵 → (𝐴𝐵) ≼ 𝐵)
24 cdadom1 8863 . . . . . . . . 9 ((𝐴𝐵) ≼ 𝐵 → ((𝐴𝐵) +𝑐 𝐵) ≼ (𝐵 +𝑐 𝐵))
2523, 24syl 17 . . . . . . . 8 ((𝐴𝐵) ≺ 𝐵 → ((𝐴𝐵) +𝑐 𝐵) ≼ (𝐵 +𝑐 𝐵))
26 domtr 7867 . . . . . . . . . . 11 ((𝐴 ≼ ((𝐴𝐵) +𝑐 𝐵) ∧ ((𝐴𝐵) +𝑐 𝐵) ≼ (𝐵 +𝑐 𝐵)) → 𝐴 ≼ (𝐵 +𝑐 𝐵))
2726ex 448 . . . . . . . . . 10 (𝐴 ≼ ((𝐴𝐵) +𝑐 𝐵) → (((𝐴𝐵) +𝑐 𝐵) ≼ (𝐵 +𝑐 𝐵) → 𝐴 ≼ (𝐵 +𝑐 𝐵)))
2821, 27syl 17 . . . . . . . . 9 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (((𝐴𝐵) +𝑐 𝐵) ≼ (𝐵 +𝑐 𝐵) → 𝐴 ≼ (𝐵 +𝑐 𝐵)))
29 simp2 1054 . . . . . . . . . . . 12 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ω ≼ 𝐴)
30 domtr 7867 . . . . . . . . . . . . 13 ((ω ≼ 𝐴𝐴 ≼ (𝐵 +𝑐 𝐵)) → ω ≼ (𝐵 +𝑐 𝐵))
3130ex 448 . . . . . . . . . . . 12 (ω ≼ 𝐴 → (𝐴 ≼ (𝐵 +𝑐 𝐵) → ω ≼ (𝐵 +𝑐 𝐵)))
3229, 31syl 17 . . . . . . . . . . 11 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴 ≼ (𝐵 +𝑐 𝐵) → ω ≼ (𝐵 +𝑐 𝐵)))
33 cdainf 8869 . . . . . . . . . . . . 13 (ω ≼ 𝐵 ↔ ω ≼ (𝐵 +𝑐 𝐵))
3433biimpri 216 . . . . . . . . . . . 12 (ω ≼ (𝐵 +𝑐 𝐵) → ω ≼ 𝐵)
35 domrefg 7848 . . . . . . . . . . . . 13 (𝐵 ∈ dom card → 𝐵𝐵)
36 infcdaabs 8883 . . . . . . . . . . . . . . 15 ((𝐵 ∈ dom card ∧ ω ≼ 𝐵𝐵𝐵) → (𝐵 +𝑐 𝐵) ≈ 𝐵)
37363com23 1262 . . . . . . . . . . . . . 14 ((𝐵 ∈ dom card ∧ 𝐵𝐵 ∧ ω ≼ 𝐵) → (𝐵 +𝑐 𝐵) ≈ 𝐵)
38373expia 1258 . . . . . . . . . . . . 13 ((𝐵 ∈ dom card ∧ 𝐵𝐵) → (ω ≼ 𝐵 → (𝐵 +𝑐 𝐵) ≈ 𝐵))
3935, 38mpdan 698 . . . . . . . . . . . 12 (𝐵 ∈ dom card → (ω ≼ 𝐵 → (𝐵 +𝑐 𝐵) ≈ 𝐵))
408, 34, 39syl2im 39 . . . . . . . . . . 11 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (ω ≼ (𝐵 +𝑐 𝐵) → (𝐵 +𝑐 𝐵) ≈ 𝐵))
4132, 40syld 45 . . . . . . . . . 10 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴 ≼ (𝐵 +𝑐 𝐵) → (𝐵 +𝑐 𝐵) ≈ 𝐵))
42 domen2 7960 . . . . . . . . . . 11 ((𝐵 +𝑐 𝐵) ≈ 𝐵 → (𝐴 ≼ (𝐵 +𝑐 𝐵) ↔ 𝐴𝐵))
4342biimpcd 237 . . . . . . . . . 10 (𝐴 ≼ (𝐵 +𝑐 𝐵) → ((𝐵 +𝑐 𝐵) ≈ 𝐵𝐴𝐵))
4441, 43sylcom 30 . . . . . . . . 9 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴 ≼ (𝐵 +𝑐 𝐵) → 𝐴𝐵))
4528, 44syld 45 . . . . . . . 8 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (((𝐴𝐵) +𝑐 𝐵) ≼ (𝐵 +𝑐 𝐵) → 𝐴𝐵))
46 domnsym 7943 . . . . . . . 8 (𝐴𝐵 → ¬ 𝐵𝐴)
4725, 45, 46syl56 35 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ((𝐴𝐵) ≺ 𝐵 → ¬ 𝐵𝐴))
4822, 47mt2d 129 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ¬ (𝐴𝐵) ≺ 𝐵)
49 domtri2 8670 . . . . . . 7 ((𝐵 ∈ dom card ∧ (𝐴𝐵) ∈ dom card) → (𝐵 ≼ (𝐴𝐵) ↔ ¬ (𝐴𝐵) ≺ 𝐵))
508, 16, 49syl2anc 690 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐵 ≼ (𝐴𝐵) ↔ ¬ (𝐴𝐵) ≺ 𝐵))
5148, 50mpbird 245 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐵 ≼ (𝐴𝐵))
52 cdadom2 8864 . . . . 5 (𝐵 ≼ (𝐴𝐵) → ((𝐴𝐵) +𝑐 𝐵) ≼ ((𝐴𝐵) +𝑐 (𝐴𝐵)))
5351, 52syl 17 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ((𝐴𝐵) +𝑐 𝐵) ≼ ((𝐴𝐵) +𝑐 (𝐴𝐵)))
54 domtr 7867 . . . 4 ((𝐴 ≼ ((𝐴𝐵) +𝑐 𝐵) ∧ ((𝐴𝐵) +𝑐 𝐵) ≼ ((𝐴𝐵) +𝑐 (𝐴𝐵))) → 𝐴 ≼ ((𝐴𝐵) +𝑐 (𝐴𝐵)))
5521, 53, 54syl2anc 690 . . 3 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐴 ≼ ((𝐴𝐵) +𝑐 (𝐴𝐵)))
56 domtr 7867 . . . . . 6 ((ω ≼ 𝐴𝐴 ≼ ((𝐴𝐵) +𝑐 (𝐴𝐵))) → ω ≼ ((𝐴𝐵) +𝑐 (𝐴𝐵)))
5729, 55, 56syl2anc 690 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ω ≼ ((𝐴𝐵) +𝑐 (𝐴𝐵)))
58 cdainf 8869 . . . . 5 (ω ≼ (𝐴𝐵) ↔ ω ≼ ((𝐴𝐵) +𝑐 (𝐴𝐵)))
5957, 58sylibr 222 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ω ≼ (𝐴𝐵))
60 domrefg 7848 . . . . 5 ((𝐴𝐵) ∈ dom card → (𝐴𝐵) ≼ (𝐴𝐵))
6116, 60syl 17 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≼ (𝐴𝐵))
62 infcdaabs 8883 . . . 4 (((𝐴𝐵) ∈ dom card ∧ ω ≼ (𝐴𝐵) ∧ (𝐴𝐵) ≼ (𝐴𝐵)) → ((𝐴𝐵) +𝑐 (𝐴𝐵)) ≈ (𝐴𝐵))
6316, 59, 61, 62syl3anc 1317 . . 3 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ((𝐴𝐵) +𝑐 (𝐴𝐵)) ≈ (𝐴𝐵))
64 domentr 7873 . . 3 ((𝐴 ≼ ((𝐴𝐵) +𝑐 (𝐴𝐵)) ∧ ((𝐴𝐵) +𝑐 (𝐴𝐵)) ≈ (𝐴𝐵)) → 𝐴 ≼ (𝐴𝐵))
6555, 63, 64syl2anc 690 . 2 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐴 ≼ (𝐴𝐵))
66 sbth 7937 . 2 (((𝐴𝐵) ≼ 𝐴𝐴 ≼ (𝐴𝐵)) → (𝐴𝐵) ≈ 𝐴)
674, 65, 66syl2anc 690 1 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  w3a 1030  wcel 1975  cdif 3531  cun 3532  wss 3534   class class class wbr 4572  dom cdm 5023  (class class class)co 6522  ωcom 6929  cen 7810  cdom 7811  csdm 7812  cardccrd 8616   +𝑐 ccda 8844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-rep 4688  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819  ax-inf2 8393
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-ral 2895  df-rex 2896  df-reu 2897  df-rmo 2898  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-pss 3550  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-tp 4124  df-op 4126  df-uni 4362  df-int 4400  df-iun 4446  df-br 4573  df-opab 4633  df-mpt 4634  df-tr 4670  df-eprel 4934  df-id 4938  df-po 4944  df-so 4945  df-fr 4982  df-se 4983  df-we 4984  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-pred 5578  df-ord 5624  df-on 5625  df-lim 5626  df-suc 5627  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-isom 5794  df-riota 6484  df-ov 6525  df-oprab 6526  df-mpt2 6527  df-om 6930  df-1st 7031  df-2nd 7032  df-wrecs 7266  df-recs 7327  df-rdg 7365  df-1o 7419  df-2o 7420  df-oadd 7423  df-er 7601  df-en 7814  df-dom 7815  df-sdom 7816  df-fin 7817  df-oi 8270  df-card 8620  df-cda 8845
This theorem is referenced by:  infdif2  8887  alephsuc3  9253  aleph1irr  14755
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