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Theorem infdif 10204
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 1137 . . 3 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐴 ∈ dom card)
2 difss 4132 . . 3 (𝐴𝐵) ⊆ 𝐴
3 ssdomg 8996 . . 3 (𝐴 ∈ dom card → ((𝐴𝐵) ⊆ 𝐴 → (𝐴𝐵) ≼ 𝐴))
41, 2, 3mpisyl 21 . 2 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≼ 𝐴)
5 sdomdom 8976 . . . . . . . . 9 (𝐵𝐴𝐵𝐴)
653ad2ant3 1136 . . . . . . . 8 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐵𝐴)
7 numdom 10033 . . . . . . . 8 ((𝐴 ∈ dom card ∧ 𝐵𝐴) → 𝐵 ∈ dom card)
81, 6, 7syl2anc 585 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐵 ∈ dom card)
9 unnum 10191 . . . . . . 7 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵) ∈ dom card)
101, 8, 9syl2anc 585 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ∈ dom card)
11 ssun1 4173 . . . . . 6 𝐴 ⊆ (𝐴𝐵)
12 ssdomg 8996 . . . . . 6 ((𝐴𝐵) ∈ dom card → (𝐴 ⊆ (𝐴𝐵) → 𝐴 ≼ (𝐴𝐵)))
1310, 11, 12mpisyl 21 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐴 ≼ (𝐴𝐵))
14 undif1 4476 . . . . . 6 ((𝐴𝐵) ∪ 𝐵) = (𝐴𝐵)
15 ssnum 10034 . . . . . . . 8 ((𝐴 ∈ dom card ∧ (𝐴𝐵) ⊆ 𝐴) → (𝐴𝐵) ∈ dom card)
161, 2, 15sylancl 587 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ∈ dom card)
17 undjudom 10162 . . . . . . 7 (((𝐴𝐵) ∈ dom card ∧ 𝐵 ∈ dom card) → ((𝐴𝐵) ∪ 𝐵) ≼ ((𝐴𝐵) ⊔ 𝐵))
1816, 8, 17syl2anc 585 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ((𝐴𝐵) ∪ 𝐵) ≼ ((𝐴𝐵) ⊔ 𝐵))
1914, 18eqbrtrrid 5185 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≼ ((𝐴𝐵) ⊔ 𝐵))
20 domtr 9003 . . . . 5 ((𝐴 ≼ (𝐴𝐵) ∧ (𝐴𝐵) ≼ ((𝐴𝐵) ⊔ 𝐵)) → 𝐴 ≼ ((𝐴𝐵) ⊔ 𝐵))
2113, 19, 20syl2anc 585 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐴 ≼ ((𝐴𝐵) ⊔ 𝐵))
22 simp3 1139 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐵𝐴)
23 sdomdom 8976 . . . . . . . . 9 ((𝐴𝐵) ≺ 𝐵 → (𝐴𝐵) ≼ 𝐵)
24 relsdom 8946 . . . . . . . . . 10 Rel ≺
2524brrelex2i 5734 . . . . . . . . 9 ((𝐴𝐵) ≺ 𝐵𝐵 ∈ V)
26 djudom1 10177 . . . . . . . . 9 (((𝐴𝐵) ≼ 𝐵𝐵 ∈ V) → ((𝐴𝐵) ⊔ 𝐵) ≼ (𝐵𝐵))
2723, 25, 26syl2anc 585 . . . . . . . 8 ((𝐴𝐵) ≺ 𝐵 → ((𝐴𝐵) ⊔ 𝐵) ≼ (𝐵𝐵))
28 domtr 9003 . . . . . . . . . . 11 ((𝐴 ≼ ((𝐴𝐵) ⊔ 𝐵) ∧ ((𝐴𝐵) ⊔ 𝐵) ≼ (𝐵𝐵)) → 𝐴 ≼ (𝐵𝐵))
2928ex 414 . . . . . . . . . 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 1138 . . . . . . . . . . . 12 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ω ≼ 𝐴)
32 domtr 9003 . . . . . . . . . . . . 13 ((ω ≼ 𝐴𝐴 ≼ (𝐵𝐵)) → ω ≼ (𝐵𝐵))
3332ex 414 . . . . . . . . . . . 12 (ω ≼ 𝐴 → (𝐴 ≼ (𝐵𝐵) → ω ≼ (𝐵𝐵)))
3431, 33syl 17 . . . . . . . . . . 11 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴 ≼ (𝐵𝐵) → ω ≼ (𝐵𝐵)))
35 djuinf 10183 . . . . . . . . . . . . 13 (ω ≼ 𝐵 ↔ ω ≼ (𝐵𝐵))
3635biimpri 227 . . . . . . . . . . . 12 (ω ≼ (𝐵𝐵) → ω ≼ 𝐵)
37 domrefg 8983 . . . . . . . . . . . . 13 (𝐵 ∈ dom card → 𝐵𝐵)
38 infdjuabs 10201 . . . . . . . . . . . . . . 15 ((𝐵 ∈ dom card ∧ ω ≼ 𝐵𝐵𝐵) → (𝐵𝐵) ≈ 𝐵)
39383com23 1127 . . . . . . . . . . . . . 14 ((𝐵 ∈ dom card ∧ 𝐵𝐵 ∧ ω ≼ 𝐵) → (𝐵𝐵) ≈ 𝐵)
40393expia 1122 . . . . . . . . . . . . 13 ((𝐵 ∈ dom card ∧ 𝐵𝐵) → (ω ≼ 𝐵 → (𝐵𝐵) ≈ 𝐵))
4137, 40mpdan 686 . . . . . . . . . . . 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 9120 . . . . . . . . . . 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 9099 . . . . . . . 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 9984 . . . . . . 7 ((𝐵 ∈ dom card ∧ (𝐴𝐵) ∈ dom card) → (𝐵 ≼ (𝐴𝐵) ↔ ¬ (𝐴𝐵) ≺ 𝐵))
528, 16, 51syl2anc 585 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐵 ≼ (𝐴𝐵) ↔ ¬ (𝐴𝐵) ≺ 𝐵))
5350, 52mpbird 257 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐵 ≼ (𝐴𝐵))
541difexd 5330 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ∈ V)
55 djudom2 10178 . . . . 5 ((𝐵 ≼ (𝐴𝐵) ∧ (𝐴𝐵) ∈ V) → ((𝐴𝐵) ⊔ 𝐵) ≼ ((𝐴𝐵) ⊔ (𝐴𝐵)))
5653, 54, 55syl2anc 585 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ((𝐴𝐵) ⊔ 𝐵) ≼ ((𝐴𝐵) ⊔ (𝐴𝐵)))
57 domtr 9003 . . . 4 ((𝐴 ≼ ((𝐴𝐵) ⊔ 𝐵) ∧ ((𝐴𝐵) ⊔ 𝐵) ≼ ((𝐴𝐵) ⊔ (𝐴𝐵))) → 𝐴 ≼ ((𝐴𝐵) ⊔ (𝐴𝐵)))
5821, 56, 57syl2anc 585 . . 3 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐴 ≼ ((𝐴𝐵) ⊔ (𝐴𝐵)))
59 domtr 9003 . . . . . 6 ((ω ≼ 𝐴𝐴 ≼ ((𝐴𝐵) ⊔ (𝐴𝐵))) → ω ≼ ((𝐴𝐵) ⊔ (𝐴𝐵)))
6031, 58, 59syl2anc 585 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ω ≼ ((𝐴𝐵) ⊔ (𝐴𝐵)))
61 djuinf 10183 . . . . 5 (ω ≼ (𝐴𝐵) ↔ ω ≼ ((𝐴𝐵) ⊔ (𝐴𝐵)))
6260, 61sylibr 233 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ω ≼ (𝐴𝐵))
63 domrefg 8983 . . . . 5 ((𝐴𝐵) ∈ dom card → (𝐴𝐵) ≼ (𝐴𝐵))
6416, 63syl 17 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≼ (𝐴𝐵))
65 infdjuabs 10201 . . . 4 (((𝐴𝐵) ∈ dom card ∧ ω ≼ (𝐴𝐵) ∧ (𝐴𝐵) ≼ (𝐴𝐵)) → ((𝐴𝐵) ⊔ (𝐴𝐵)) ≈ (𝐴𝐵))
6616, 62, 64, 65syl3anc 1372 . . 3 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ((𝐴𝐵) ⊔ (𝐴𝐵)) ≈ (𝐴𝐵))
67 domentr 9009 . . 3 ((𝐴 ≼ ((𝐴𝐵) ⊔ (𝐴𝐵)) ∧ ((𝐴𝐵) ⊔ (𝐴𝐵)) ≈ (𝐴𝐵)) → 𝐴 ≼ (𝐴𝐵))
6858, 66, 67syl2anc 585 . 2 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐴 ≼ (𝐴𝐵))
69 sbth 9093 . 2 (((𝐴𝐵) ≼ 𝐴𝐴 ≼ (𝐴𝐵)) → (𝐴𝐵) ≈ 𝐴)
704, 68, 69syl2anc 585 1 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  w3a 1088  wcel 2107  Vcvv 3475  cdif 3946  cun 3947  wss 3949   class class class wbr 5149  dom cdm 5677  ωcom 7855  cen 8936  cdom 8937  csdm 8938  cdju 9893  cardccrd 9930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-2o 8467  df-oadd 8470  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-oi 9505  df-dju 9896  df-card 9934
This theorem is referenced by:  infdif2  10205  alephsuc3  10575  aleph1irr  16189
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