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Theorem infdif 9016
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 1059 . . 3 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐴 ∈ dom card)
2 difss 3729 . . 3 (𝐴𝐵) ⊆ 𝐴
3 ssdomg 7986 . . 3 (𝐴 ∈ dom card → ((𝐴𝐵) ⊆ 𝐴 → (𝐴𝐵) ≼ 𝐴))
41, 2, 3mpisyl 21 . 2 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≼ 𝐴)
5 sdomdom 7968 . . . . . . . . 9 (𝐵𝐴𝐵𝐴)
653ad2ant3 1082 . . . . . . . 8 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐵𝐴)
7 numdom 8846 . . . . . . . 8 ((𝐴 ∈ dom card ∧ 𝐵𝐴) → 𝐵 ∈ dom card)
81, 6, 7syl2anc 692 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐵 ∈ dom card)
9 unnum 9007 . . . . . . 7 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵) ∈ dom card)
101, 8, 9syl2anc 692 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ∈ dom card)
11 ssun1 3768 . . . . . 6 𝐴 ⊆ (𝐴𝐵)
12 ssdomg 7986 . . . . . 6 ((𝐴𝐵) ∈ dom card → (𝐴 ⊆ (𝐴𝐵) → 𝐴 ≼ (𝐴𝐵)))
1310, 11, 12mpisyl 21 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐴 ≼ (𝐴𝐵))
14 undif1 4034 . . . . . 6 ((𝐴𝐵) ∪ 𝐵) = (𝐴𝐵)
15 ssnum 8847 . . . . . . . 8 ((𝐴 ∈ dom card ∧ (𝐴𝐵) ⊆ 𝐴) → (𝐴𝐵) ∈ dom card)
161, 2, 15sylancl 693 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ∈ dom card)
17 uncdadom 8978 . . . . . . 7 (((𝐴𝐵) ∈ dom card ∧ 𝐵 ∈ dom card) → ((𝐴𝐵) ∪ 𝐵) ≼ ((𝐴𝐵) +𝑐 𝐵))
1816, 8, 17syl2anc 692 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ((𝐴𝐵) ∪ 𝐵) ≼ ((𝐴𝐵) +𝑐 𝐵))
1914, 18syl5eqbrr 4680 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≼ ((𝐴𝐵) +𝑐 𝐵))
20 domtr 7994 . . . . 5 ((𝐴 ≼ (𝐴𝐵) ∧ (𝐴𝐵) ≼ ((𝐴𝐵) +𝑐 𝐵)) → 𝐴 ≼ ((𝐴𝐵) +𝑐 𝐵))
2113, 19, 20syl2anc 692 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐴 ≼ ((𝐴𝐵) +𝑐 𝐵))
22 simp3 1061 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐵𝐴)
23 sdomdom 7968 . . . . . . . . 9 ((𝐴𝐵) ≺ 𝐵 → (𝐴𝐵) ≼ 𝐵)
24 cdadom1 8993 . . . . . . . . 9 ((𝐴𝐵) ≼ 𝐵 → ((𝐴𝐵) +𝑐 𝐵) ≼ (𝐵 +𝑐 𝐵))
2523, 24syl 17 . . . . . . . 8 ((𝐴𝐵) ≺ 𝐵 → ((𝐴𝐵) +𝑐 𝐵) ≼ (𝐵 +𝑐 𝐵))
26 domtr 7994 . . . . . . . . . . 11 ((𝐴 ≼ ((𝐴𝐵) +𝑐 𝐵) ∧ ((𝐴𝐵) +𝑐 𝐵) ≼ (𝐵 +𝑐 𝐵)) → 𝐴 ≼ (𝐵 +𝑐 𝐵))
2726ex 450 . . . . . . . . . 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 1060 . . . . . . . . . . . 12 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ω ≼ 𝐴)
30 domtr 7994 . . . . . . . . . . . . 13 ((ω ≼ 𝐴𝐴 ≼ (𝐵 +𝑐 𝐵)) → ω ≼ (𝐵 +𝑐 𝐵))
3130ex 450 . . . . . . . . . . . 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 8999 . . . . . . . . . . . . 13 (ω ≼ 𝐵 ↔ ω ≼ (𝐵 +𝑐 𝐵))
3433biimpri 218 . . . . . . . . . . . 12 (ω ≼ (𝐵 +𝑐 𝐵) → ω ≼ 𝐵)
35 domrefg 7975 . . . . . . . . . . . . 13 (𝐵 ∈ dom card → 𝐵𝐵)
36 infcdaabs 9013 . . . . . . . . . . . . . . 15 ((𝐵 ∈ dom card ∧ ω ≼ 𝐵𝐵𝐵) → (𝐵 +𝑐 𝐵) ≈ 𝐵)
37363com23 1269 . . . . . . . . . . . . . 14 ((𝐵 ∈ dom card ∧ 𝐵𝐵 ∧ ω ≼ 𝐵) → (𝐵 +𝑐 𝐵) ≈ 𝐵)
38373expia 1265 . . . . . . . . . . . . 13 ((𝐵 ∈ dom card ∧ 𝐵𝐵) → (ω ≼ 𝐵 → (𝐵 +𝑐 𝐵) ≈ 𝐵))
3935, 38mpdan 701 . . . . . . . . . . . 12 (𝐵 ∈ dom card → (ω ≼ 𝐵 → (𝐵 +𝑐 𝐵) ≈ 𝐵))
408, 34, 39syl2im 40 . . . . . . . . . . 11 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (ω ≼ (𝐵 +𝑐 𝐵) → (𝐵 +𝑐 𝐵) ≈ 𝐵))
4132, 40syld 47 . . . . . . . . . 10 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴 ≼ (𝐵 +𝑐 𝐵) → (𝐵 +𝑐 𝐵) ≈ 𝐵))
42 domen2 8088 . . . . . . . . . . 11 ((𝐵 +𝑐 𝐵) ≈ 𝐵 → (𝐴 ≼ (𝐵 +𝑐 𝐵) ↔ 𝐴𝐵))
4342biimpcd 239 . . . . . . . . . 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 47 . . . . . . . 8 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (((𝐴𝐵) +𝑐 𝐵) ≼ (𝐵 +𝑐 𝐵) → 𝐴𝐵))
46 domnsym 8071 . . . . . . . 8 (𝐴𝐵 → ¬ 𝐵𝐴)
4725, 45, 46syl56 36 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ((𝐴𝐵) ≺ 𝐵 → ¬ 𝐵𝐴))
4822, 47mt2d 131 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ¬ (𝐴𝐵) ≺ 𝐵)
49 domtri2 8800 . . . . . . 7 ((𝐵 ∈ dom card ∧ (𝐴𝐵) ∈ dom card) → (𝐵 ≼ (𝐴𝐵) ↔ ¬ (𝐴𝐵) ≺ 𝐵))
508, 16, 49syl2anc 692 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐵 ≼ (𝐴𝐵) ↔ ¬ (𝐴𝐵) ≺ 𝐵))
5148, 50mpbird 247 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐵 ≼ (𝐴𝐵))
52 cdadom2 8994 . . . . 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 7994 . . . 4 ((𝐴 ≼ ((𝐴𝐵) +𝑐 𝐵) ∧ ((𝐴𝐵) +𝑐 𝐵) ≼ ((𝐴𝐵) +𝑐 (𝐴𝐵))) → 𝐴 ≼ ((𝐴𝐵) +𝑐 (𝐴𝐵)))
5521, 53, 54syl2anc 692 . . 3 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐴 ≼ ((𝐴𝐵) +𝑐 (𝐴𝐵)))
56 domtr 7994 . . . . . 6 ((ω ≼ 𝐴𝐴 ≼ ((𝐴𝐵) +𝑐 (𝐴𝐵))) → ω ≼ ((𝐴𝐵) +𝑐 (𝐴𝐵)))
5729, 55, 56syl2anc 692 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ω ≼ ((𝐴𝐵) +𝑐 (𝐴𝐵)))
58 cdainf 8999 . . . . 5 (ω ≼ (𝐴𝐵) ↔ ω ≼ ((𝐴𝐵) +𝑐 (𝐴𝐵)))
5957, 58sylibr 224 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ω ≼ (𝐴𝐵))
60 domrefg 7975 . . . . 5 ((𝐴𝐵) ∈ dom card → (𝐴𝐵) ≼ (𝐴𝐵))
6116, 60syl 17 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≼ (𝐴𝐵))
62 infcdaabs 9013 . . . 4 (((𝐴𝐵) ∈ dom card ∧ ω ≼ (𝐴𝐵) ∧ (𝐴𝐵) ≼ (𝐴𝐵)) → ((𝐴𝐵) +𝑐 (𝐴𝐵)) ≈ (𝐴𝐵))
6316, 59, 61, 62syl3anc 1324 . . 3 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ((𝐴𝐵) +𝑐 (𝐴𝐵)) ≈ (𝐴𝐵))
64 domentr 8000 . . 3 ((𝐴 ≼ ((𝐴𝐵) +𝑐 (𝐴𝐵)) ∧ ((𝐴𝐵) +𝑐 (𝐴𝐵)) ≈ (𝐴𝐵)) → 𝐴 ≼ (𝐴𝐵))
6555, 63, 64syl2anc 692 . 2 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐴 ≼ (𝐴𝐵))
66 sbth 8065 . 2 (((𝐴𝐵) ≼ 𝐴𝐴 ≼ (𝐴𝐵)) → (𝐴𝐵) ≈ 𝐴)
674, 65, 66syl2anc 692 1 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  w3a 1036  wcel 1988  cdif 3564  cun 3565  wss 3567   class class class wbr 4644  dom cdm 5104  (class class class)co 6635  ωcom 7050  cen 7937  cdom 7938  csdm 7939  cardccrd 8746   +𝑐 ccda 8974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-2o 7546  df-oadd 7549  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-oi 8400  df-card 8750  df-cda 8975
This theorem is referenced by:  infdif2  9017  alephsuc3  9387  aleph1irr  14956
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