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Theorem infdiffi 8499
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 3700 . . . . . 6 (𝑥 = ∅ → (𝐴𝑥) = (𝐴 ∖ ∅))
2 dif0 3924 . . . . . 6 (𝐴 ∖ ∅) = 𝐴
31, 2syl6eq 2671 . . . . 5 (𝑥 = ∅ → (𝐴𝑥) = 𝐴)
43breq1d 4623 . . . 4 (𝑥 = ∅ → ((𝐴𝑥) ≈ 𝐴𝐴𝐴))
54imbi2d 330 . . 3 (𝑥 = ∅ → ((ω ≼ 𝐴 → (𝐴𝑥) ≈ 𝐴) ↔ (ω ≼ 𝐴𝐴𝐴)))
6 difeq2 3700 . . . . 5 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴𝑦))
76breq1d 4623 . . . 4 (𝑥 = 𝑦 → ((𝐴𝑥) ≈ 𝐴 ↔ (𝐴𝑦) ≈ 𝐴))
87imbi2d 330 . . 3 (𝑥 = 𝑦 → ((ω ≼ 𝐴 → (𝐴𝑥) ≈ 𝐴) ↔ (ω ≼ 𝐴 → (𝐴𝑦) ≈ 𝐴)))
9 difeq2 3700 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → (𝐴𝑥) = (𝐴 ∖ (𝑦 ∪ {𝑧})))
10 difun1 3863 . . . . . 6 (𝐴 ∖ (𝑦 ∪ {𝑧})) = ((𝐴𝑦) ∖ {𝑧})
119, 10syl6eq 2671 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → (𝐴𝑥) = ((𝐴𝑦) ∖ {𝑧}))
1211breq1d 4623 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐴𝑥) ≈ 𝐴 ↔ ((𝐴𝑦) ∖ {𝑧}) ≈ 𝐴))
1312imbi2d 330 . . 3 (𝑥 = (𝑦 ∪ {𝑧}) → ((ω ≼ 𝐴 → (𝐴𝑥) ≈ 𝐴) ↔ (ω ≼ 𝐴 → ((𝐴𝑦) ∖ {𝑧}) ≈ 𝐴)))
14 difeq2 3700 . . . . 5 (𝑥 = 𝐵 → (𝐴𝑥) = (𝐴𝐵))
1514breq1d 4623 . . . 4 (𝑥 = 𝐵 → ((𝐴𝑥) ≈ 𝐴 ↔ (𝐴𝐵) ≈ 𝐴))
1615imbi2d 330 . . 3 (𝑥 = 𝐵 → ((ω ≼ 𝐴 → (𝐴𝑥) ≈ 𝐴) ↔ (ω ≼ 𝐴 → (𝐴𝐵) ≈ 𝐴)))
17 reldom 7905 . . . . 5 Rel ≼
1817brrelex2i 5119 . . . 4 (ω ≼ 𝐴𝐴 ∈ V)
19 enrefg 7931 . . . 4 (𝐴 ∈ V → 𝐴𝐴)
2018, 19syl 17 . . 3 (ω ≼ 𝐴𝐴𝐴)
21 domen2 8047 . . . . . . . . 9 ((𝐴𝑦) ≈ 𝐴 → (ω ≼ (𝐴𝑦) ↔ ω ≼ 𝐴))
2221biimparc 504 . . . . . . . 8 ((ω ≼ 𝐴 ∧ (𝐴𝑦) ≈ 𝐴) → ω ≼ (𝐴𝑦))
23 infdifsn 8498 . . . . . . . 8 (ω ≼ (𝐴𝑦) → ((𝐴𝑦) ∖ {𝑧}) ≈ (𝐴𝑦))
2422, 23syl 17 . . . . . . 7 ((ω ≼ 𝐴 ∧ (𝐴𝑦) ≈ 𝐴) → ((𝐴𝑦) ∖ {𝑧}) ≈ (𝐴𝑦))
25 entr 7952 . . . . . . 7 ((((𝐴𝑦) ∖ {𝑧}) ≈ (𝐴𝑦) ∧ (𝐴𝑦) ≈ 𝐴) → ((𝐴𝑦) ∖ {𝑧}) ≈ 𝐴)
2624, 25sylancom 700 . . . . . 6 ((ω ≼ 𝐴 ∧ (𝐴𝑦) ≈ 𝐴) → ((𝐴𝑦) ∖ {𝑧}) ≈ 𝐴)
2726ex 450 . . . . 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 8144 . 2 (𝐵 ∈ Fin → (ω ≼ 𝐴 → (𝐴𝐵) ≈ 𝐴))
3130impcom 446 1 ((ω ≼ 𝐴𝐵 ∈ Fin) → (𝐴𝐵) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  cdif 3552  cun 3553  c0 3891  {csn 4148   class class class wbr 4613  ωcom 7012  cen 7896  cdom 7897  Fincfn 7899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-om 7013  df-1o 7505  df-er 7687  df-en 7900  df-dom 7901  df-fin 7903
This theorem is referenced by: (None)
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