MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  infdif Structured version   Visualization version   GIF version

Theorem infdif 10186
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 1136 . . 3 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐴 ∈ dom card)
2 difss 4127 . . 3 (𝐴𝐵) ⊆ 𝐴
3 ssdomg 8979 . . 3 (𝐴 ∈ dom card → ((𝐴𝐵) ⊆ 𝐴 → (𝐴𝐵) ≼ 𝐴))
41, 2, 3mpisyl 21 . 2 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≼ 𝐴)
5 sdomdom 8959 . . . . . . . . 9 (𝐵𝐴𝐵𝐴)
653ad2ant3 1135 . . . . . . . 8 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐵𝐴)
7 numdom 10015 . . . . . . . 8 ((𝐴 ∈ dom card ∧ 𝐵𝐴) → 𝐵 ∈ dom card)
81, 6, 7syl2anc 584 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐵 ∈ dom card)
9 unnum 10173 . . . . . . 7 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵) ∈ dom card)
101, 8, 9syl2anc 584 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ∈ dom card)
11 ssun1 4168 . . . . . 6 𝐴 ⊆ (𝐴𝐵)
12 ssdomg 8979 . . . . . 6 ((𝐴𝐵) ∈ dom card → (𝐴 ⊆ (𝐴𝐵) → 𝐴 ≼ (𝐴𝐵)))
1310, 11, 12mpisyl 21 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐴 ≼ (𝐴𝐵))
14 undif1 4471 . . . . . 6 ((𝐴𝐵) ∪ 𝐵) = (𝐴𝐵)
15 ssnum 10016 . . . . . . . 8 ((𝐴 ∈ dom card ∧ (𝐴𝐵) ⊆ 𝐴) → (𝐴𝐵) ∈ dom card)
161, 2, 15sylancl 586 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ∈ dom card)
17 undjudom 10144 . . . . . . 7 (((𝐴𝐵) ∈ dom card ∧ 𝐵 ∈ dom card) → ((𝐴𝐵) ∪ 𝐵) ≼ ((𝐴𝐵) ⊔ 𝐵))
1816, 8, 17syl2anc 584 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ((𝐴𝐵) ∪ 𝐵) ≼ ((𝐴𝐵) ⊔ 𝐵))
1914, 18eqbrtrrid 5177 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≼ ((𝐴𝐵) ⊔ 𝐵))
20 domtr 8986 . . . . 5 ((𝐴 ≼ (𝐴𝐵) ∧ (𝐴𝐵) ≼ ((𝐴𝐵) ⊔ 𝐵)) → 𝐴 ≼ ((𝐴𝐵) ⊔ 𝐵))
2113, 19, 20syl2anc 584 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐴 ≼ ((𝐴𝐵) ⊔ 𝐵))
22 simp3 1138 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐵𝐴)
23 sdomdom 8959 . . . . . . . . 9 ((𝐴𝐵) ≺ 𝐵 → (𝐴𝐵) ≼ 𝐵)
24 relsdom 8929 . . . . . . . . . 10 Rel ≺
2524brrelex2i 5725 . . . . . . . . 9 ((𝐴𝐵) ≺ 𝐵𝐵 ∈ V)
26 djudom1 10159 . . . . . . . . 9 (((𝐴𝐵) ≼ 𝐵𝐵 ∈ V) → ((𝐴𝐵) ⊔ 𝐵) ≼ (𝐵𝐵))
2723, 25, 26syl2anc 584 . . . . . . . 8 ((𝐴𝐵) ≺ 𝐵 → ((𝐴𝐵) ⊔ 𝐵) ≼ (𝐵𝐵))
28 domtr 8986 . . . . . . . . . . 11 ((𝐴 ≼ ((𝐴𝐵) ⊔ 𝐵) ∧ ((𝐴𝐵) ⊔ 𝐵) ≼ (𝐵𝐵)) → 𝐴 ≼ (𝐵𝐵))
2928ex 413 . . . . . . . . . 10 (𝐴 ≼ ((𝐴𝐵) ⊔ 𝐵) → (((𝐴𝐵) ⊔ 𝐵) ≼ (𝐵𝐵) → 𝐴 ≼ (𝐵𝐵)))
3021, 29syl 17 . . . . . . . . 9 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (((𝐴𝐵) ⊔ 𝐵) ≼ (𝐵𝐵) → 𝐴 ≼ (𝐵𝐵)))
31 simp2 1137 . . . . . . . . . . . 12 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ω ≼ 𝐴)
32 domtr 8986 . . . . . . . . . . . . 13 ((ω ≼ 𝐴𝐴 ≼ (𝐵𝐵)) → ω ≼ (𝐵𝐵))
3332ex 413 . . . . . . . . . . . 12 (ω ≼ 𝐴 → (𝐴 ≼ (𝐵𝐵) → ω ≼ (𝐵𝐵)))
3431, 33syl 17 . . . . . . . . . . 11 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴 ≼ (𝐵𝐵) → ω ≼ (𝐵𝐵)))
35 djuinf 10165 . . . . . . . . . . . . 13 (ω ≼ 𝐵 ↔ ω ≼ (𝐵𝐵))
3635biimpri 227 . . . . . . . . . . . 12 (ω ≼ (𝐵𝐵) → ω ≼ 𝐵)
37 domrefg 8966 . . . . . . . . . . . . 13 (𝐵 ∈ dom card → 𝐵𝐵)
38 infdjuabs 10183 . . . . . . . . . . . . . . 15 ((𝐵 ∈ dom card ∧ ω ≼ 𝐵𝐵𝐵) → (𝐵𝐵) ≈ 𝐵)
39383com23 1126 . . . . . . . . . . . . . 14 ((𝐵 ∈ dom card ∧ 𝐵𝐵 ∧ ω ≼ 𝐵) → (𝐵𝐵) ≈ 𝐵)
40393expia 1121 . . . . . . . . . . . . 13 ((𝐵 ∈ dom card ∧ 𝐵𝐵) → (ω ≼ 𝐵 → (𝐵𝐵) ≈ 𝐵))
4137, 40mpdan 685 . . . . . . . . . . . 12 (𝐵 ∈ dom card → (ω ≼ 𝐵 → (𝐵𝐵) ≈ 𝐵))
428, 36, 41syl2im 40 . . . . . . . . . . 11 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (ω ≼ (𝐵𝐵) → (𝐵𝐵) ≈ 𝐵))
4334, 42syld 47 . . . . . . . . . 10 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴 ≼ (𝐵𝐵) → (𝐵𝐵) ≈ 𝐵))
44 domen2 9103 . . . . . . . . . . 11 ((𝐵𝐵) ≈ 𝐵 → (𝐴 ≼ (𝐵𝐵) ↔ 𝐴𝐵))
4544biimpcd 248 . . . . . . . . . 10 (𝐴 ≼ (𝐵𝐵) → ((𝐵𝐵) ≈ 𝐵𝐴𝐵))
4643, 45sylcom 30 . . . . . . . . 9 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴 ≼ (𝐵𝐵) → 𝐴𝐵))
4730, 46syld 47 . . . . . . . 8 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (((𝐴𝐵) ⊔ 𝐵) ≼ (𝐵𝐵) → 𝐴𝐵))
48 domnsym 9082 . . . . . . . 8 (𝐴𝐵 → ¬ 𝐵𝐴)
4927, 47, 48syl56 36 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ((𝐴𝐵) ≺ 𝐵 → ¬ 𝐵𝐴))
5022, 49mt2d 136 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ¬ (𝐴𝐵) ≺ 𝐵)
51 domtri2 9966 . . . . . . 7 ((𝐵 ∈ dom card ∧ (𝐴𝐵) ∈ dom card) → (𝐵 ≼ (𝐴𝐵) ↔ ¬ (𝐴𝐵) ≺ 𝐵))
528, 16, 51syl2anc 584 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐵 ≼ (𝐴𝐵) ↔ ¬ (𝐴𝐵) ≺ 𝐵))
5350, 52mpbird 256 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐵 ≼ (𝐴𝐵))
541difexd 5322 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ∈ V)
55 djudom2 10160 . . . . 5 ((𝐵 ≼ (𝐴𝐵) ∧ (𝐴𝐵) ∈ V) → ((𝐴𝐵) ⊔ 𝐵) ≼ ((𝐴𝐵) ⊔ (𝐴𝐵)))
5653, 54, 55syl2anc 584 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ((𝐴𝐵) ⊔ 𝐵) ≼ ((𝐴𝐵) ⊔ (𝐴𝐵)))
57 domtr 8986 . . . 4 ((𝐴 ≼ ((𝐴𝐵) ⊔ 𝐵) ∧ ((𝐴𝐵) ⊔ 𝐵) ≼ ((𝐴𝐵) ⊔ (𝐴𝐵))) → 𝐴 ≼ ((𝐴𝐵) ⊔ (𝐴𝐵)))
5821, 56, 57syl2anc 584 . . 3 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐴 ≼ ((𝐴𝐵) ⊔ (𝐴𝐵)))
59 domtr 8986 . . . . . 6 ((ω ≼ 𝐴𝐴 ≼ ((𝐴𝐵) ⊔ (𝐴𝐵))) → ω ≼ ((𝐴𝐵) ⊔ (𝐴𝐵)))
6031, 58, 59syl2anc 584 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ω ≼ ((𝐴𝐵) ⊔ (𝐴𝐵)))
61 djuinf 10165 . . . . 5 (ω ≼ (𝐴𝐵) ↔ ω ≼ ((𝐴𝐵) ⊔ (𝐴𝐵)))
6260, 61sylibr 233 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ω ≼ (𝐴𝐵))
63 domrefg 8966 . . . . 5 ((𝐴𝐵) ∈ dom card → (𝐴𝐵) ≼ (𝐴𝐵))
6416, 63syl 17 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≼ (𝐴𝐵))
65 infdjuabs 10183 . . . 4 (((𝐴𝐵) ∈ dom card ∧ ω ≼ (𝐴𝐵) ∧ (𝐴𝐵) ≼ (𝐴𝐵)) → ((𝐴𝐵) ⊔ (𝐴𝐵)) ≈ (𝐴𝐵))
6616, 62, 64, 65syl3anc 1371 . . 3 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ((𝐴𝐵) ⊔ (𝐴𝐵)) ≈ (𝐴𝐵))
67 domentr 8992 . . 3 ((𝐴 ≼ ((𝐴𝐵) ⊔ (𝐴𝐵)) ∧ ((𝐴𝐵) ⊔ (𝐴𝐵)) ≈ (𝐴𝐵)) → 𝐴 ≼ (𝐴𝐵))
6858, 66, 67syl2anc 584 . 2 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐴 ≼ (𝐴𝐵))
69 sbth 9076 . 2 (((𝐴𝐵) ≼ 𝐴𝐴 ≼ (𝐴𝐵)) → (𝐴𝐵) ≈ 𝐴)
704, 68, 69syl2anc 584 1 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  w3a 1087  wcel 2106  Vcvv 3473  cdif 3941  cun 3942  wss 3944   class class class wbr 5141  dom cdm 5669  ωcom 7838  cen 8919  cdom 8920  csdm 8921  cdju 9875  cardccrd 9912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708  ax-inf2 9618
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-isom 6541  df-riota 7349  df-ov 7396  df-oprab 7397  df-mpo 7398  df-om 7839  df-1st 7957  df-2nd 7958  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-1o 8448  df-2o 8449  df-oadd 8452  df-er 8686  df-en 8923  df-dom 8924  df-sdom 8925  df-fin 8926  df-oi 9487  df-dju 9878  df-card 9916
This theorem is referenced by:  infdif2  10187  alephsuc3  10557  aleph1irr  16171
  Copyright terms: Public domain W3C validator