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Theorem infdiffi 9416
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 4051 . . . . . 6 (𝑥 = ∅ → (𝐴𝑥) = (𝐴 ∖ ∅))
2 dif0 4306 . . . . . 6 (𝐴 ∖ ∅) = 𝐴
31, 2eqtrdi 2794 . . . . 5 (𝑥 = ∅ → (𝐴𝑥) = 𝐴)
43breq1d 5084 . . . 4 (𝑥 = ∅ → ((𝐴𝑥) ≈ 𝐴𝐴𝐴))
54imbi2d 341 . . 3 (𝑥 = ∅ → ((ω ≼ 𝐴 → (𝐴𝑥) ≈ 𝐴) ↔ (ω ≼ 𝐴𝐴𝐴)))
6 difeq2 4051 . . . . 5 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴𝑦))
76breq1d 5084 . . . 4 (𝑥 = 𝑦 → ((𝐴𝑥) ≈ 𝐴 ↔ (𝐴𝑦) ≈ 𝐴))
87imbi2d 341 . . 3 (𝑥 = 𝑦 → ((ω ≼ 𝐴 → (𝐴𝑥) ≈ 𝐴) ↔ (ω ≼ 𝐴 → (𝐴𝑦) ≈ 𝐴)))
9 difeq2 4051 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → (𝐴𝑥) = (𝐴 ∖ (𝑦 ∪ {𝑧})))
10 difun1 4223 . . . . . 6 (𝐴 ∖ (𝑦 ∪ {𝑧})) = ((𝐴𝑦) ∖ {𝑧})
119, 10eqtrdi 2794 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → (𝐴𝑥) = ((𝐴𝑦) ∖ {𝑧}))
1211breq1d 5084 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐴𝑥) ≈ 𝐴 ↔ ((𝐴𝑦) ∖ {𝑧}) ≈ 𝐴))
1312imbi2d 341 . . 3 (𝑥 = (𝑦 ∪ {𝑧}) → ((ω ≼ 𝐴 → (𝐴𝑥) ≈ 𝐴) ↔ (ω ≼ 𝐴 → ((𝐴𝑦) ∖ {𝑧}) ≈ 𝐴)))
14 difeq2 4051 . . . . 5 (𝑥 = 𝐵 → (𝐴𝑥) = (𝐴𝐵))
1514breq1d 5084 . . . 4 (𝑥 = 𝐵 → ((𝐴𝑥) ≈ 𝐴 ↔ (𝐴𝐵) ≈ 𝐴))
1615imbi2d 341 . . 3 (𝑥 = 𝐵 → ((ω ≼ 𝐴 → (𝐴𝑥) ≈ 𝐴) ↔ (ω ≼ 𝐴 → (𝐴𝐵) ≈ 𝐴)))
17 reldom 8739 . . . . 5 Rel ≼
1817brrelex2i 5644 . . . 4 (ω ≼ 𝐴𝐴 ∈ V)
19 enrefg 8772 . . . 4 (𝐴 ∈ V → 𝐴𝐴)
2018, 19syl 17 . . 3 (ω ≼ 𝐴𝐴𝐴)
21 domen2 8907 . . . . . . . . 9 ((𝐴𝑦) ≈ 𝐴 → (ω ≼ (𝐴𝑦) ↔ ω ≼ 𝐴))
2221biimparc 480 . . . . . . . 8 ((ω ≼ 𝐴 ∧ (𝐴𝑦) ≈ 𝐴) → ω ≼ (𝐴𝑦))
23 infdifsn 9415 . . . . . . . 8 (ω ≼ (𝐴𝑦) → ((𝐴𝑦) ∖ {𝑧}) ≈ (𝐴𝑦))
2422, 23syl 17 . . . . . . 7 ((ω ≼ 𝐴 ∧ (𝐴𝑦) ≈ 𝐴) → ((𝐴𝑦) ∖ {𝑧}) ≈ (𝐴𝑦))
25 entr 8792 . . . . . . 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 8947 . 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 1539  wcel 2106  Vcvv 3432  cdif 3884  cun 3885  c0 4256  {csn 4561   class class class wbr 5074  ωcom 7712  cen 8730  cdom 8731  Fincfn 8733
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-om 7713  df-er 8498  df-en 8734  df-dom 8735  df-fin 8737
This theorem is referenced by: (None)
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