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Theorem infdif 10249
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 4135 . . 3 (𝐴𝐵) ⊆ 𝐴
3 ssdomg 9041 . . 3 (𝐴 ∈ dom card → ((𝐴𝐵) ⊆ 𝐴 → (𝐴𝐵) ≼ 𝐴))
41, 2, 3mpisyl 21 . 2 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≼ 𝐴)
5 sdomdom 9021 . . . . . . . . 9 (𝐵𝐴𝐵𝐴)
653ad2ant3 1135 . . . . . . . 8 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐵𝐴)
7 numdom 10079 . . . . . . . 8 ((𝐴 ∈ dom card ∧ 𝐵𝐴) → 𝐵 ∈ dom card)
81, 6, 7syl2anc 584 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐵 ∈ dom card)
9 unnum 10238 . . . . . . 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 4177 . . . . . 6 𝐴 ⊆ (𝐴𝐵)
12 ssdomg 9041 . . . . . 6 ((𝐴𝐵) ∈ dom card → (𝐴 ⊆ (𝐴𝐵) → 𝐴 ≼ (𝐴𝐵)))
1310, 11, 12mpisyl 21 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐴 ≼ (𝐴𝐵))
14 undif1 4475 . . . . . 6 ((𝐴𝐵) ∪ 𝐵) = (𝐴𝐵)
15 ssnum 10080 . . . . . . . 8 ((𝐴 ∈ dom card ∧ (𝐴𝐵) ⊆ 𝐴) → (𝐴𝐵) ∈ dom card)
161, 2, 15sylancl 586 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ∈ dom card)
17 undjudom 10209 . . . . . . 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 5178 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≼ ((𝐴𝐵) ⊔ 𝐵))
20 domtr 9048 . . . . 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 9021 . . . . . . . . 9 ((𝐴𝐵) ≺ 𝐵 → (𝐴𝐵) ≼ 𝐵)
24 relsdom 8993 . . . . . . . . . 10 Rel ≺
2524brrelex2i 5741 . . . . . . . . 9 ((𝐴𝐵) ≺ 𝐵𝐵 ∈ V)
26 djudom1 10224 . . . . . . . . 9 (((𝐴𝐵) ≼ 𝐵𝐵 ∈ V) → ((𝐴𝐵) ⊔ 𝐵) ≼ (𝐵𝐵))
2723, 25, 26syl2anc 584 . . . . . . . 8 ((𝐴𝐵) ≺ 𝐵 → ((𝐴𝐵) ⊔ 𝐵) ≼ (𝐵𝐵))
28 domtr 9048 . . . . . . . . . . 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 1137 . . . . . . . . . . . 12 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ω ≼ 𝐴)
32 domtr 9048 . . . . . . . . . . . . 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 10230 . . . . . . . . . . . . 13 (ω ≼ 𝐵 ↔ ω ≼ (𝐵𝐵))
3635biimpri 228 . . . . . . . . . . . 12 (ω ≼ (𝐵𝐵) → ω ≼ 𝐵)
37 domrefg 9028 . . . . . . . . . . . . 13 (𝐵 ∈ dom card → 𝐵𝐵)
38 infdjuabs 10246 . . . . . . . . . . . . . . 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 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 9161 . . . . . . . . . . 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 9140 . . . . . . . 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 10030 . . . . . . 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 5330 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ∈ V)
55 djudom2 10225 . . . . 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 9048 . . . 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 9048 . . . . . 6 ((ω ≼ 𝐴𝐴 ≼ ((𝐴𝐵) ⊔ (𝐴𝐵))) → ω ≼ ((𝐴𝐵) ⊔ (𝐴𝐵)))
6031, 58, 59syl2anc 584 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ω ≼ ((𝐴𝐵) ⊔ (𝐴𝐵)))
61 djuinf 10230 . . . . 5 (ω ≼ (𝐴𝐵) ↔ ω ≼ ((𝐴𝐵) ⊔ (𝐴𝐵)))
6260, 61sylibr 234 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ω ≼ (𝐴𝐵))
63 domrefg 9028 . . . . 5 ((𝐴𝐵) ∈ dom card → (𝐴𝐵) ≼ (𝐴𝐵))
6416, 63syl 17 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≼ (𝐴𝐵))
65 infdjuabs 10246 . . . 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 9054 . . 3 ((𝐴 ≼ ((𝐴𝐵) ⊔ (𝐴𝐵)) ∧ ((𝐴𝐵) ⊔ (𝐴𝐵)) ≈ (𝐴𝐵)) → 𝐴 ≼ (𝐴𝐵))
6858, 66, 67syl2anc 584 . 2 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐴 ≼ (𝐴𝐵))
69 sbth 9134 . 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 2107  Vcvv 3479  cdif 3947  cun 3948  wss 3950   class class class wbr 5142  dom cdm 5684  ωcom 7888  cen 8983  cdom 8984  csdm 8985  cdju 9939  cardccrd 9976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-inf2 9682
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-oadd 8511  df-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-oi 9551  df-dju 9942  df-card 9980
This theorem is referenced by:  infdif2  10250  alephsuc3  10621  aleph1irr  16283
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