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Theorem infdiffi 9723
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 4137 . . . . . 6 (𝑥 = ∅ → (𝐴𝑥) = (𝐴 ∖ ∅))
2 dif0 4395 . . . . . 6 (𝐴 ∖ ∅) = 𝐴
31, 2eqtrdi 2790 . . . . 5 (𝑥 = ∅ → (𝐴𝑥) = 𝐴)
43breq1d 5179 . . . 4 (𝑥 = ∅ → ((𝐴𝑥) ≈ 𝐴𝐴𝐴))
54imbi2d 340 . . 3 (𝑥 = ∅ → ((ω ≼ 𝐴 → (𝐴𝑥) ≈ 𝐴) ↔ (ω ≼ 𝐴𝐴𝐴)))
6 difeq2 4137 . . . . 5 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴𝑦))
76breq1d 5179 . . . 4 (𝑥 = 𝑦 → ((𝐴𝑥) ≈ 𝐴 ↔ (𝐴𝑦) ≈ 𝐴))
87imbi2d 340 . . 3 (𝑥 = 𝑦 → ((ω ≼ 𝐴 → (𝐴𝑥) ≈ 𝐴) ↔ (ω ≼ 𝐴 → (𝐴𝑦) ≈ 𝐴)))
9 difeq2 4137 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → (𝐴𝑥) = (𝐴 ∖ (𝑦 ∪ {𝑧})))
10 difun1 4313 . . . . . 6 (𝐴 ∖ (𝑦 ∪ {𝑧})) = ((𝐴𝑦) ∖ {𝑧})
119, 10eqtrdi 2790 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → (𝐴𝑥) = ((𝐴𝑦) ∖ {𝑧}))
1211breq1d 5179 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐴𝑥) ≈ 𝐴 ↔ ((𝐴𝑦) ∖ {𝑧}) ≈ 𝐴))
1312imbi2d 340 . . 3 (𝑥 = (𝑦 ∪ {𝑧}) → ((ω ≼ 𝐴 → (𝐴𝑥) ≈ 𝐴) ↔ (ω ≼ 𝐴 → ((𝐴𝑦) ∖ {𝑧}) ≈ 𝐴)))
14 difeq2 4137 . . . . 5 (𝑥 = 𝐵 → (𝐴𝑥) = (𝐴𝐵))
1514breq1d 5179 . . . 4 (𝑥 = 𝐵 → ((𝐴𝑥) ≈ 𝐴 ↔ (𝐴𝐵) ≈ 𝐴))
1615imbi2d 340 . . 3 (𝑥 = 𝐵 → ((ω ≼ 𝐴 → (𝐴𝑥) ≈ 𝐴) ↔ (ω ≼ 𝐴 → (𝐴𝐵) ≈ 𝐴)))
17 reldom 9005 . . . . 5 Rel ≼
1817brrelex2i 5756 . . . 4 (ω ≼ 𝐴𝐴 ∈ V)
19 enrefg 9040 . . . 4 (𝐴 ∈ V → 𝐴𝐴)
2018, 19syl 17 . . 3 (ω ≼ 𝐴𝐴𝐴)
21 domen2 9182 . . . . . . . . 9 ((𝐴𝑦) ≈ 𝐴 → (ω ≼ (𝐴𝑦) ↔ ω ≼ 𝐴))
2221biimparc 479 . . . . . . . 8 ((ω ≼ 𝐴 ∧ (𝐴𝑦) ≈ 𝐴) → ω ≼ (𝐴𝑦))
23 infdifsn 9722 . . . . . . . 8 (ω ≼ (𝐴𝑦) → ((𝐴𝑦) ∖ {𝑧}) ≈ (𝐴𝑦))
2422, 23syl 17 . . . . . . 7 ((ω ≼ 𝐴 ∧ (𝐴𝑦) ≈ 𝐴) → ((𝐴𝑦) ∖ {𝑧}) ≈ (𝐴𝑦))
25 entr 9062 . . . . . . 7 ((((𝐴𝑦) ∖ {𝑧}) ≈ (𝐴𝑦) ∧ (𝐴𝑦) ≈ 𝐴) → ((𝐴𝑦) ∖ {𝑧}) ≈ 𝐴)
2624, 25sylancom 587 . . . . . 6 ((ω ≼ 𝐴 ∧ (𝐴𝑦) ≈ 𝐴) → ((𝐴𝑦) ∖ {𝑧}) ≈ 𝐴)
2726ex 412 . . . . 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 9226 . 2 (𝐵 ∈ Fin → (ω ≼ 𝐴 → (𝐴𝐵) ≈ 𝐴))
3130impcom 407 1 ((ω ≼ 𝐴𝐵 ∈ Fin) → (𝐴𝐵) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2103  Vcvv 3482  cdif 3967  cun 3968  c0 4347  {csn 4648   class class class wbr 5169  ωcom 7899  cen 8996  cdom 8997  Fincfn 8999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-pss 3990  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5170  df-opab 5232  df-mpt 5253  df-tr 5287  df-id 5597  df-eprel 5603  df-po 5611  df-so 5612  df-fr 5654  df-we 5656  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-ord 6397  df-on 6398  df-lim 6399  df-suc 6400  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-om 7900  df-er 8759  df-en 9000  df-dom 9001  df-fin 9003
This theorem is referenced by: (None)
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