Theorem infdiffi 9653

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 ≼ 𝐴 ∧ 𝐵 ∈ Fin) → (𝐴 ∖ 𝐵) ≈ 𝐴)

Proof of Theorem infdiffi

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		difeq2 4117	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐴 ∖ 𝑥) = (𝐴 ∖ ∅))
2		dif0 4373	. . . . . 6 ⊢ (𝐴 ∖ ∅) = 𝐴
3	1, 2	eqtrdi 2789	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 ∖ 𝑥) = 𝐴)
4	3	breq1d 5159	. . . 4 ⊢ (𝑥 = ∅ → ((𝐴 ∖ 𝑥) ≈ 𝐴 ↔ 𝐴 ≈ 𝐴))
5	4	imbi2d 341	. . 3 ⊢ (𝑥 = ∅ → ((ω ≼ 𝐴 → (𝐴 ∖ 𝑥) ≈ 𝐴) ↔ (ω ≼ 𝐴 → 𝐴 ≈ 𝐴)))
6		difeq2 4117	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝑦))
7	6	breq1d 5159	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 ∖ 𝑥) ≈ 𝐴 ↔ (𝐴 ∖ 𝑦) ≈ 𝐴))
8	7	imbi2d 341	. . 3 ⊢ (𝑥 = 𝑦 → ((ω ≼ 𝐴 → (𝐴 ∖ 𝑥) ≈ 𝐴) ↔ (ω ≼ 𝐴 → (𝐴 ∖ 𝑦) ≈ 𝐴)))
9		difeq2 4117	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝐴 ∖ 𝑥) = (𝐴 ∖ (𝑦 ∪ {𝑧})))
10		difun1 4290	. . . . . 6 ⊢ (𝐴 ∖ (𝑦 ∪ {𝑧})) = ((𝐴 ∖ 𝑦) ∖ {𝑧})
11	9, 10	eqtrdi 2789	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝐴 ∖ 𝑥) = ((𝐴 ∖ 𝑦) ∖ {𝑧}))
12	11	breq1d 5159	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐴 ∖ 𝑥) ≈ 𝐴 ↔ ((𝐴 ∖ 𝑦) ∖ {𝑧}) ≈ 𝐴))
13	12	imbi2d 341	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((ω ≼ 𝐴 → (𝐴 ∖ 𝑥) ≈ 𝐴) ↔ (ω ≼ 𝐴 → ((𝐴 ∖ 𝑦) ∖ {𝑧}) ≈ 𝐴)))
14		difeq2 4117	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝐵))
15	14	breq1d 5159	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝐴 ∖ 𝑥) ≈ 𝐴 ↔ (𝐴 ∖ 𝐵) ≈ 𝐴))
16	15	imbi2d 341	. . 3 ⊢ (𝑥 = 𝐵 → ((ω ≼ 𝐴 → (𝐴 ∖ 𝑥) ≈ 𝐴) ↔ (ω ≼ 𝐴 → (𝐴 ∖ 𝐵) ≈ 𝐴)))
17		reldom 8945	. . . . 5 ⊢ Rel ≼
18	17	brrelex2i 5734	. . . 4 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V)
19		enrefg 8980	. . . 4 ⊢ (𝐴 ∈ V → 𝐴 ≈ 𝐴)
20	18, 19	syl 17	. . 3 ⊢ (ω ≼ 𝐴 → 𝐴 ≈ 𝐴)
21		domen2 9120	. . . . . . . . 9 ⊢ ((𝐴 ∖ 𝑦) ≈ 𝐴 → (ω ≼ (𝐴 ∖ 𝑦) ↔ ω ≼ 𝐴))
22	21	biimparc 481	. . . . . . . 8 ⊢ ((ω ≼ 𝐴 ∧ (𝐴 ∖ 𝑦) ≈ 𝐴) → ω ≼ (𝐴 ∖ 𝑦))
23		infdifsn 9652	. . . . . . . 8 ⊢ (ω ≼ (𝐴 ∖ 𝑦) → ((𝐴 ∖ 𝑦) ∖ {𝑧}) ≈ (𝐴 ∖ 𝑦))
24	22, 23	syl 17	. . . . . . 7 ⊢ ((ω ≼ 𝐴 ∧ (𝐴 ∖ 𝑦) ≈ 𝐴) → ((𝐴 ∖ 𝑦) ∖ {𝑧}) ≈ (𝐴 ∖ 𝑦))
25		entr 9002	. . . . . . 7 ⊢ ((((𝐴 ∖ 𝑦) ∖ {𝑧}) ≈ (𝐴 ∖ 𝑦) ∧ (𝐴 ∖ 𝑦) ≈ 𝐴) → ((𝐴 ∖ 𝑦) ∖ {𝑧}) ≈ 𝐴)
26	24, 25	sylancom 589	. . . . . 6 ⊢ ((ω ≼ 𝐴 ∧ (𝐴 ∖ 𝑦) ≈ 𝐴) → ((𝐴 ∖ 𝑦) ∖ {𝑧}) ≈ 𝐴)
27	26	ex 414	. . . . 5 ⊢ (ω ≼ 𝐴 → ((𝐴 ∖ 𝑦) ≈ 𝐴 → ((𝐴 ∖ 𝑦) ∖ {𝑧}) ≈ 𝐴))
28	27	a2i 14	. . . 4 ⊢ ((ω ≼ 𝐴 → (𝐴 ∖ 𝑦) ≈ 𝐴) → (ω ≼ 𝐴 → ((𝐴 ∖ 𝑦) ∖ {𝑧}) ≈ 𝐴))
29	28	a1i 11	. . 3 ⊢ (𝑦 ∈ Fin → ((ω ≼ 𝐴 → (𝐴 ∖ 𝑦) ≈ 𝐴) → (ω ≼ 𝐴 → ((𝐴 ∖ 𝑦) ∖ {𝑧}) ≈ 𝐴)))
30	5, 8, 13, 16, 20, 29	findcard2 9164	. 2 ⊢ (𝐵 ∈ Fin → (ω ≼ 𝐴 → (𝐴 ∖ 𝐵) ≈ 𝐴))
31	30	impcom 409	1 ⊢ ((ω ≼ 𝐴 ∧ 𝐵 ∈ Fin) → (𝐴 ∖ 𝐵) ≈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3475   ∖ cdif 3946   ∪ cun 3947  ∅c0 4323  {csn 4629   class class class wbr 5149  ωcom 7855   ≈ cen 8936   ≼ cdom 8937  Fincfn 8939

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-om 7856  df-er 8703  df-en 8940  df-dom 8941  df-fin 8943

This theorem is referenced by: (None)