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Theorem infdiffi 9109
Description: Removing a finite set from an infinite set does not change the cardinality of the set. (Contributed by Mario Carneiro, 30-Apr-2015.)
Assertion
Ref Expression
infdiffi ((ω ≼ 𝐴𝐵 ∈ Fin) → (𝐴𝐵) ≈ 𝐴)

Proof of Theorem infdiffi
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 difeq2 4090 . . . . . 6 (𝑥 = ∅ → (𝐴𝑥) = (𝐴 ∖ ∅))
2 dif0 4329 . . . . . 6 (𝐴 ∖ ∅) = 𝐴
31, 2syl6eq 2869 . . . . 5 (𝑥 = ∅ → (𝐴𝑥) = 𝐴)
43breq1d 5067 . . . 4 (𝑥 = ∅ → ((𝐴𝑥) ≈ 𝐴𝐴𝐴))
54imbi2d 342 . . 3 (𝑥 = ∅ → ((ω ≼ 𝐴 → (𝐴𝑥) ≈ 𝐴) ↔ (ω ≼ 𝐴𝐴𝐴)))
6 difeq2 4090 . . . . 5 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴𝑦))
76breq1d 5067 . . . 4 (𝑥 = 𝑦 → ((𝐴𝑥) ≈ 𝐴 ↔ (𝐴𝑦) ≈ 𝐴))
87imbi2d 342 . . 3 (𝑥 = 𝑦 → ((ω ≼ 𝐴 → (𝐴𝑥) ≈ 𝐴) ↔ (ω ≼ 𝐴 → (𝐴𝑦) ≈ 𝐴)))
9 difeq2 4090 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → (𝐴𝑥) = (𝐴 ∖ (𝑦 ∪ {𝑧})))
10 difun1 4261 . . . . . 6 (𝐴 ∖ (𝑦 ∪ {𝑧})) = ((𝐴𝑦) ∖ {𝑧})
119, 10syl6eq 2869 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → (𝐴𝑥) = ((𝐴𝑦) ∖ {𝑧}))
1211breq1d 5067 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐴𝑥) ≈ 𝐴 ↔ ((𝐴𝑦) ∖ {𝑧}) ≈ 𝐴))
1312imbi2d 342 . . 3 (𝑥 = (𝑦 ∪ {𝑧}) → ((ω ≼ 𝐴 → (𝐴𝑥) ≈ 𝐴) ↔ (ω ≼ 𝐴 → ((𝐴𝑦) ∖ {𝑧}) ≈ 𝐴)))
14 difeq2 4090 . . . . 5 (𝑥 = 𝐵 → (𝐴𝑥) = (𝐴𝐵))
1514breq1d 5067 . . . 4 (𝑥 = 𝐵 → ((𝐴𝑥) ≈ 𝐴 ↔ (𝐴𝐵) ≈ 𝐴))
1615imbi2d 342 . . 3 (𝑥 = 𝐵 → ((ω ≼ 𝐴 → (𝐴𝑥) ≈ 𝐴) ↔ (ω ≼ 𝐴 → (𝐴𝐵) ≈ 𝐴)))
17 reldom 8503 . . . . 5 Rel ≼
1817brrelex2i 5602 . . . 4 (ω ≼ 𝐴𝐴 ∈ V)
19 enrefg 8529 . . . 4 (𝐴 ∈ V → 𝐴𝐴)
2018, 19syl 17 . . 3 (ω ≼ 𝐴𝐴𝐴)
21 domen2 8648 . . . . . . . . 9 ((𝐴𝑦) ≈ 𝐴 → (ω ≼ (𝐴𝑦) ↔ ω ≼ 𝐴))
2221biimparc 480 . . . . . . . 8 ((ω ≼ 𝐴 ∧ (𝐴𝑦) ≈ 𝐴) → ω ≼ (𝐴𝑦))
23 infdifsn 9108 . . . . . . . 8 (ω ≼ (𝐴𝑦) → ((𝐴𝑦) ∖ {𝑧}) ≈ (𝐴𝑦))
2422, 23syl 17 . . . . . . 7 ((ω ≼ 𝐴 ∧ (𝐴𝑦) ≈ 𝐴) → ((𝐴𝑦) ∖ {𝑧}) ≈ (𝐴𝑦))
25 entr 8549 . . . . . . 7 ((((𝐴𝑦) ∖ {𝑧}) ≈ (𝐴𝑦) ∧ (𝐴𝑦) ≈ 𝐴) → ((𝐴𝑦) ∖ {𝑧}) ≈ 𝐴)
2624, 25sylancom 588 . . . . . 6 ((ω ≼ 𝐴 ∧ (𝐴𝑦) ≈ 𝐴) → ((𝐴𝑦) ∖ {𝑧}) ≈ 𝐴)
2726ex 413 . . . . 5 (ω ≼ 𝐴 → ((𝐴𝑦) ≈ 𝐴 → ((𝐴𝑦) ∖ {𝑧}) ≈ 𝐴))
2827a2i 14 . . . 4 ((ω ≼ 𝐴 → (𝐴𝑦) ≈ 𝐴) → (ω ≼ 𝐴 → ((𝐴𝑦) ∖ {𝑧}) ≈ 𝐴))
2928a1i 11 . . 3 (𝑦 ∈ Fin → ((ω ≼ 𝐴 → (𝐴𝑦) ≈ 𝐴) → (ω ≼ 𝐴 → ((𝐴𝑦) ∖ {𝑧}) ≈ 𝐴)))
305, 8, 13, 16, 20, 29findcard2 8746 . 2 (𝐵 ∈ Fin → (ω ≼ 𝐴 → (𝐴𝐵) ≈ 𝐴))
3130impcom 408 1 ((ω ≼ 𝐴𝐵 ∈ Fin) → (𝐴𝐵) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  Vcvv 3492  cdif 3930  cun 3931  c0 4288  {csn 4557   class class class wbr 5057  ωcom 7569  cen 8494  cdom 8495  Fincfn 8497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-om 7570  df-1o 8091  df-er 8278  df-en 8498  df-dom 8499  df-fin 8501
This theorem is referenced by: (None)
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