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Theorem infdif 9823
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 1138 . . 3 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐴 ∈ dom card)
2 difss 4046 . . 3 (𝐴𝐵) ⊆ 𝐴
3 ssdomg 8674 . . 3 (𝐴 ∈ dom card → ((𝐴𝐵) ⊆ 𝐴 → (𝐴𝐵) ≼ 𝐴))
41, 2, 3mpisyl 21 . 2 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≼ 𝐴)
5 sdomdom 8656 . . . . . . . . 9 (𝐵𝐴𝐵𝐴)
653ad2ant3 1137 . . . . . . . 8 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐵𝐴)
7 numdom 9652 . . . . . . . 8 ((𝐴 ∈ dom card ∧ 𝐵𝐴) → 𝐵 ∈ dom card)
81, 6, 7syl2anc 587 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐵 ∈ dom card)
9 unnum 9810 . . . . . . 7 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵) ∈ dom card)
101, 8, 9syl2anc 587 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ∈ dom card)
11 ssun1 4086 . . . . . 6 𝐴 ⊆ (𝐴𝐵)
12 ssdomg 8674 . . . . . 6 ((𝐴𝐵) ∈ dom card → (𝐴 ⊆ (𝐴𝐵) → 𝐴 ≼ (𝐴𝐵)))
1310, 11, 12mpisyl 21 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐴 ≼ (𝐴𝐵))
14 undif1 4390 . . . . . 6 ((𝐴𝐵) ∪ 𝐵) = (𝐴𝐵)
15 ssnum 9653 . . . . . . . 8 ((𝐴 ∈ dom card ∧ (𝐴𝐵) ⊆ 𝐴) → (𝐴𝐵) ∈ dom card)
161, 2, 15sylancl 589 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ∈ dom card)
17 undjudom 9781 . . . . . . 7 (((𝐴𝐵) ∈ dom card ∧ 𝐵 ∈ dom card) → ((𝐴𝐵) ∪ 𝐵) ≼ ((𝐴𝐵) ⊔ 𝐵))
1816, 8, 17syl2anc 587 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ((𝐴𝐵) ∪ 𝐵) ≼ ((𝐴𝐵) ⊔ 𝐵))
1914, 18eqbrtrrid 5089 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≼ ((𝐴𝐵) ⊔ 𝐵))
20 domtr 8681 . . . . 5 ((𝐴 ≼ (𝐴𝐵) ∧ (𝐴𝐵) ≼ ((𝐴𝐵) ⊔ 𝐵)) → 𝐴 ≼ ((𝐴𝐵) ⊔ 𝐵))
2113, 19, 20syl2anc 587 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐴 ≼ ((𝐴𝐵) ⊔ 𝐵))
22 simp3 1140 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐵𝐴)
23 sdomdom 8656 . . . . . . . . 9 ((𝐴𝐵) ≺ 𝐵 → (𝐴𝐵) ≼ 𝐵)
24 relsdom 8633 . . . . . . . . . 10 Rel ≺
2524brrelex2i 5606 . . . . . . . . 9 ((𝐴𝐵) ≺ 𝐵𝐵 ∈ V)
26 djudom1 9796 . . . . . . . . 9 (((𝐴𝐵) ≼ 𝐵𝐵 ∈ V) → ((𝐴𝐵) ⊔ 𝐵) ≼ (𝐵𝐵))
2723, 25, 26syl2anc 587 . . . . . . . 8 ((𝐴𝐵) ≺ 𝐵 → ((𝐴𝐵) ⊔ 𝐵) ≼ (𝐵𝐵))
28 domtr 8681 . . . . . . . . . . 11 ((𝐴 ≼ ((𝐴𝐵) ⊔ 𝐵) ∧ ((𝐴𝐵) ⊔ 𝐵) ≼ (𝐵𝐵)) → 𝐴 ≼ (𝐵𝐵))
2928ex 416 . . . . . . . . . 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 1139 . . . . . . . . . . . 12 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ω ≼ 𝐴)
32 domtr 8681 . . . . . . . . . . . . 13 ((ω ≼ 𝐴𝐴 ≼ (𝐵𝐵)) → ω ≼ (𝐵𝐵))
3332ex 416 . . . . . . . . . . . 12 (ω ≼ 𝐴 → (𝐴 ≼ (𝐵𝐵) → ω ≼ (𝐵𝐵)))
3431, 33syl 17 . . . . . . . . . . 11 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴 ≼ (𝐵𝐵) → ω ≼ (𝐵𝐵)))
35 djuinf 9802 . . . . . . . . . . . . 13 (ω ≼ 𝐵 ↔ ω ≼ (𝐵𝐵))
3635biimpri 231 . . . . . . . . . . . 12 (ω ≼ (𝐵𝐵) → ω ≼ 𝐵)
37 domrefg 8663 . . . . . . . . . . . . 13 (𝐵 ∈ dom card → 𝐵𝐵)
38 infdjuabs 9820 . . . . . . . . . . . . . . 15 ((𝐵 ∈ dom card ∧ ω ≼ 𝐵𝐵𝐵) → (𝐵𝐵) ≈ 𝐵)
39383com23 1128 . . . . . . . . . . . . . 14 ((𝐵 ∈ dom card ∧ 𝐵𝐵 ∧ ω ≼ 𝐵) → (𝐵𝐵) ≈ 𝐵)
40393expia 1123 . . . . . . . . . . . . 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 8789 . . . . . . . . . . 11 ((𝐵𝐵) ≈ 𝐵 → (𝐴 ≼ (𝐵𝐵) ↔ 𝐴𝐵))
4544biimpcd 252 . . . . . . . . . 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 8772 . . . . . . . 8 (𝐴𝐵 → ¬ 𝐵𝐴)
4927, 47, 48syl56 36 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ((𝐴𝐵) ≺ 𝐵 → ¬ 𝐵𝐴))
5022, 49mt2d 138 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ¬ (𝐴𝐵) ≺ 𝐵)
51 domtri2 9605 . . . . . . 7 ((𝐵 ∈ dom card ∧ (𝐴𝐵) ∈ dom card) → (𝐵 ≼ (𝐴𝐵) ↔ ¬ (𝐴𝐵) ≺ 𝐵))
528, 16, 51syl2anc 587 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐵 ≼ (𝐴𝐵) ↔ ¬ (𝐴𝐵) ≺ 𝐵))
5350, 52mpbird 260 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐵 ≼ (𝐴𝐵))
541difexd 5222 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ∈ V)
55 djudom2 9797 . . . . 5 ((𝐵 ≼ (𝐴𝐵) ∧ (𝐴𝐵) ∈ V) → ((𝐴𝐵) ⊔ 𝐵) ≼ ((𝐴𝐵) ⊔ (𝐴𝐵)))
5653, 54, 55syl2anc 587 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ((𝐴𝐵) ⊔ 𝐵) ≼ ((𝐴𝐵) ⊔ (𝐴𝐵)))
57 domtr 8681 . . . 4 ((𝐴 ≼ ((𝐴𝐵) ⊔ 𝐵) ∧ ((𝐴𝐵) ⊔ 𝐵) ≼ ((𝐴𝐵) ⊔ (𝐴𝐵))) → 𝐴 ≼ ((𝐴𝐵) ⊔ (𝐴𝐵)))
5821, 56, 57syl2anc 587 . . 3 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐴 ≼ ((𝐴𝐵) ⊔ (𝐴𝐵)))
59 domtr 8681 . . . . . 6 ((ω ≼ 𝐴𝐴 ≼ ((𝐴𝐵) ⊔ (𝐴𝐵))) → ω ≼ ((𝐴𝐵) ⊔ (𝐴𝐵)))
6031, 58, 59syl2anc 587 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ω ≼ ((𝐴𝐵) ⊔ (𝐴𝐵)))
61 djuinf 9802 . . . . 5 (ω ≼ (𝐴𝐵) ↔ ω ≼ ((𝐴𝐵) ⊔ (𝐴𝐵)))
6260, 61sylibr 237 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ω ≼ (𝐴𝐵))
63 domrefg 8663 . . . . 5 ((𝐴𝐵) ∈ dom card → (𝐴𝐵) ≼ (𝐴𝐵))
6416, 63syl 17 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≼ (𝐴𝐵))
65 infdjuabs 9820 . . . 4 (((𝐴𝐵) ∈ dom card ∧ ω ≼ (𝐴𝐵) ∧ (𝐴𝐵) ≼ (𝐴𝐵)) → ((𝐴𝐵) ⊔ (𝐴𝐵)) ≈ (𝐴𝐵))
6616, 62, 64, 65syl3anc 1373 . . 3 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ((𝐴𝐵) ⊔ (𝐴𝐵)) ≈ (𝐴𝐵))
67 domentr 8687 . . 3 ((𝐴 ≼ ((𝐴𝐵) ⊔ (𝐴𝐵)) ∧ ((𝐴𝐵) ⊔ (𝐴𝐵)) ≈ (𝐴𝐵)) → 𝐴 ≼ (𝐴𝐵))
6858, 66, 67syl2anc 587 . 2 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐴 ≼ (𝐴𝐵))
69 sbth 8766 . 2 (((𝐴𝐵) ≼ 𝐴𝐴 ≼ (𝐴𝐵)) → (𝐴𝐵) ≈ 𝐴)
704, 68, 69syl2anc 587 1 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  w3a 1089  wcel 2110  Vcvv 3408  cdif 3863  cun 3864  wss 3866   class class class wbr 5053  dom cdm 5551  ωcom 7644  cen 8623  cdom 8624  csdm 8625  cdju 9514  cardccrd 9551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-inf2 9256
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-se 5510  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isom 6389  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-2o 8203  df-oadd 8206  df-er 8391  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-oi 9126  df-dju 9517  df-card 9555
This theorem is referenced by:  infdif2  9824  alephsuc3  10194  aleph1irr  15807
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