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

Theorem infdif 10246
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 1135 . . 3 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐴 ∈ dom card)
2 difss 4146 . . 3 (𝐴𝐵) ⊆ 𝐴
3 ssdomg 9039 . . 3 (𝐴 ∈ dom card → ((𝐴𝐵) ⊆ 𝐴 → (𝐴𝐵) ≼ 𝐴))
41, 2, 3mpisyl 21 . 2 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≼ 𝐴)
5 sdomdom 9019 . . . . . . . . 9 (𝐵𝐴𝐵𝐴)
653ad2ant3 1134 . . . . . . . 8 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐵𝐴)
7 numdom 10076 . . . . . . . 8 ((𝐴 ∈ dom card ∧ 𝐵𝐴) → 𝐵 ∈ dom card)
81, 6, 7syl2anc 584 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐵 ∈ dom card)
9 unnum 10235 . . . . . . 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 4188 . . . . . 6 𝐴 ⊆ (𝐴𝐵)
12 ssdomg 9039 . . . . . 6 ((𝐴𝐵) ∈ dom card → (𝐴 ⊆ (𝐴𝐵) → 𝐴 ≼ (𝐴𝐵)))
1310, 11, 12mpisyl 21 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐴 ≼ (𝐴𝐵))
14 undif1 4482 . . . . . 6 ((𝐴𝐵) ∪ 𝐵) = (𝐴𝐵)
15 ssnum 10077 . . . . . . . 8 ((𝐴 ∈ dom card ∧ (𝐴𝐵) ⊆ 𝐴) → (𝐴𝐵) ∈ dom card)
161, 2, 15sylancl 586 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ∈ dom card)
17 undjudom 10206 . . . . . . 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 5184 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≼ ((𝐴𝐵) ⊔ 𝐵))
20 domtr 9046 . . . . 5 ((𝐴 ≼ (𝐴𝐵) ∧ (𝐴𝐵) ≼ ((𝐴𝐵) ⊔ 𝐵)) → 𝐴 ≼ ((𝐴𝐵) ⊔ 𝐵))
2113, 19, 20syl2anc 584 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐴 ≼ ((𝐴𝐵) ⊔ 𝐵))
22 simp3 1137 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐵𝐴)
23 sdomdom 9019 . . . . . . . . 9 ((𝐴𝐵) ≺ 𝐵 → (𝐴𝐵) ≼ 𝐵)
24 relsdom 8991 . . . . . . . . . 10 Rel ≺
2524brrelex2i 5746 . . . . . . . . 9 ((𝐴𝐵) ≺ 𝐵𝐵 ∈ V)
26 djudom1 10221 . . . . . . . . 9 (((𝐴𝐵) ≼ 𝐵𝐵 ∈ V) → ((𝐴𝐵) ⊔ 𝐵) ≼ (𝐵𝐵))
2723, 25, 26syl2anc 584 . . . . . . . 8 ((𝐴𝐵) ≺ 𝐵 → ((𝐴𝐵) ⊔ 𝐵) ≼ (𝐵𝐵))
28 domtr 9046 . . . . . . . . . . 11 ((𝐴 ≼ ((𝐴𝐵) ⊔ 𝐵) ∧ ((𝐴𝐵) ⊔ 𝐵) ≼ (𝐵𝐵)) → 𝐴 ≼ (𝐵𝐵))
2928ex 412 . . . . . . . . . 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 1136 . . . . . . . . . . . 12 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ω ≼ 𝐴)
32 domtr 9046 . . . . . . . . . . . . 13 ((ω ≼ 𝐴𝐴 ≼ (𝐵𝐵)) → ω ≼ (𝐵𝐵))
3332ex 412 . . . . . . . . . . . 12 (ω ≼ 𝐴 → (𝐴 ≼ (𝐵𝐵) → ω ≼ (𝐵𝐵)))
3431, 33syl 17 . . . . . . . . . . 11 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴 ≼ (𝐵𝐵) → ω ≼ (𝐵𝐵)))
35 djuinf 10227 . . . . . . . . . . . . 13 (ω ≼ 𝐵 ↔ ω ≼ (𝐵𝐵))
3635biimpri 228 . . . . . . . . . . . 12 (ω ≼ (𝐵𝐵) → ω ≼ 𝐵)
37 domrefg 9026 . . . . . . . . . . . . 13 (𝐵 ∈ dom card → 𝐵𝐵)
38 infdjuabs 10243 . . . . . . . . . . . . . . 15 ((𝐵 ∈ dom card ∧ ω ≼ 𝐵𝐵𝐵) → (𝐵𝐵) ≈ 𝐵)
39383com23 1125 . . . . . . . . . . . . . 14 ((𝐵 ∈ dom card ∧ 𝐵𝐵 ∧ ω ≼ 𝐵) → (𝐵𝐵) ≈ 𝐵)
40393expia 1120 . . . . . . . . . . . . 13 ((𝐵 ∈ dom card ∧ 𝐵𝐵) → (ω ≼ 𝐵 → (𝐵𝐵) ≈ 𝐵))
4137, 40mpdan 687 . . . . . . . . . . . 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 9159 . . . . . . . . . . 11 ((𝐵𝐵) ≈ 𝐵 → (𝐴 ≼ (𝐵𝐵) ↔ 𝐴𝐵))
4544biimpcd 249 . . . . . . . . . 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 9138 . . . . . . . 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 10027 . . . . . . 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 257 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐵 ≼ (𝐴𝐵))
541difexd 5337 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ∈ V)
55 djudom2 10222 . . . . 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 9046 . . . 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 9046 . . . . . 6 ((ω ≼ 𝐴𝐴 ≼ ((𝐴𝐵) ⊔ (𝐴𝐵))) → ω ≼ ((𝐴𝐵) ⊔ (𝐴𝐵)))
6031, 58, 59syl2anc 584 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ω ≼ ((𝐴𝐵) ⊔ (𝐴𝐵)))
61 djuinf 10227 . . . . 5 (ω ≼ (𝐴𝐵) ↔ ω ≼ ((𝐴𝐵) ⊔ (𝐴𝐵)))
6260, 61sylibr 234 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ω ≼ (𝐴𝐵))
63 domrefg 9026 . . . . 5 ((𝐴𝐵) ∈ dom card → (𝐴𝐵) ≼ (𝐴𝐵))
6416, 63syl 17 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≼ (𝐴𝐵))
65 infdjuabs 10243 . . . 4 (((𝐴𝐵) ∈ dom card ∧ ω ≼ (𝐴𝐵) ∧ (𝐴𝐵) ≼ (𝐴𝐵)) → ((𝐴𝐵) ⊔ (𝐴𝐵)) ≈ (𝐴𝐵))
6616, 62, 64, 65syl3anc 1370 . . 3 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ((𝐴𝐵) ⊔ (𝐴𝐵)) ≈ (𝐴𝐵))
67 domentr 9052 . . 3 ((𝐴 ≼ ((𝐴𝐵) ⊔ (𝐴𝐵)) ∧ ((𝐴𝐵) ⊔ (𝐴𝐵)) ≈ (𝐴𝐵)) → 𝐴 ≼ (𝐴𝐵))
6858, 66, 67syl2anc 584 . 2 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐴 ≼ (𝐴𝐵))
69 sbth 9132 . 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 206  w3a 1086  wcel 2106  Vcvv 3478  cdif 3960  cun 3961  wss 3963   class class class wbr 5148  dom cdm 5689  ωcom 7887  cen 8981  cdom 8982  csdm 8983  cdju 9936  cardccrd 9973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-oi 9548  df-dju 9939  df-card 9977
This theorem is referenced by:  infdif2  10247  alephsuc3  10618  aleph1irr  16279
  Copyright terms: Public domain W3C validator