Theorem infdiffi 9681

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 4113	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐴 ∖ 𝑥) = (𝐴 ∖ ∅))
2		dif0 4373	. . . . . 6 ⊢ (𝐴 ∖ ∅) = 𝐴
3	1, 2	eqtrdi 2781	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 ∖ 𝑥) = 𝐴)
4	3	breq1d 5158	. . . 4 ⊢ (𝑥 = ∅ → ((𝐴 ∖ 𝑥) ≈ 𝐴 ↔ 𝐴 ≈ 𝐴))
5	4	imbi2d 339	. . 3 ⊢ (𝑥 = ∅ → ((ω ≼ 𝐴 → (𝐴 ∖ 𝑥) ≈ 𝐴) ↔ (ω ≼ 𝐴 → 𝐴 ≈ 𝐴)))
6		difeq2 4113	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝑦))
7	6	breq1d 5158	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 ∖ 𝑥) ≈ 𝐴 ↔ (𝐴 ∖ 𝑦) ≈ 𝐴))
8	7	imbi2d 339	. . 3 ⊢ (𝑥 = 𝑦 → ((ω ≼ 𝐴 → (𝐴 ∖ 𝑥) ≈ 𝐴) ↔ (ω ≼ 𝐴 → (𝐴 ∖ 𝑦) ≈ 𝐴)))
9		difeq2 4113	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝐴 ∖ 𝑥) = (𝐴 ∖ (𝑦 ∪ {𝑧})))
10		difun1 4289	. . . . . 6 ⊢ (𝐴 ∖ (𝑦 ∪ {𝑧})) = ((𝐴 ∖ 𝑦) ∖ {𝑧})
11	9, 10	eqtrdi 2781	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝐴 ∖ 𝑥) = ((𝐴 ∖ 𝑦) ∖ {𝑧}))
12	11	breq1d 5158	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐴 ∖ 𝑥) ≈ 𝐴 ↔ ((𝐴 ∖ 𝑦) ∖ {𝑧}) ≈ 𝐴))
13	12	imbi2d 339	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((ω ≼ 𝐴 → (𝐴 ∖ 𝑥) ≈ 𝐴) ↔ (ω ≼ 𝐴 → ((𝐴 ∖ 𝑦) ∖ {𝑧}) ≈ 𝐴)))
14		difeq2 4113	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝐵))
15	14	breq1d 5158	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝐴 ∖ 𝑥) ≈ 𝐴 ↔ (𝐴 ∖ 𝐵) ≈ 𝐴))
16	15	imbi2d 339	. . 3 ⊢ (𝑥 = 𝐵 → ((ω ≼ 𝐴 → (𝐴 ∖ 𝑥) ≈ 𝐴) ↔ (ω ≼ 𝐴 → (𝐴 ∖ 𝐵) ≈ 𝐴)))
17		reldom 8968	. . . . 5 ⊢ Rel ≼
18	17	brrelex2i 5734	. . . 4 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V)
19		enrefg 9003	. . . 4 ⊢ (𝐴 ∈ V → 𝐴 ≈ 𝐴)
20	18, 19	syl 17	. . 3 ⊢ (ω ≼ 𝐴 → 𝐴 ≈ 𝐴)
21		domen2 9143	. . . . . . . . 9 ⊢ ((𝐴 ∖ 𝑦) ≈ 𝐴 → (ω ≼ (𝐴 ∖ 𝑦) ↔ ω ≼ 𝐴))
22	21	biimparc 478	. . . . . . . 8 ⊢ ((ω ≼ 𝐴 ∧ (𝐴 ∖ 𝑦) ≈ 𝐴) → ω ≼ (𝐴 ∖ 𝑦))
23		infdifsn 9680	. . . . . . . 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 9025	. . . . . . 7 ⊢ ((((𝐴 ∖ 𝑦) ∖ {𝑧}) ≈ (𝐴 ∖ 𝑦) ∧ (𝐴 ∖ 𝑦) ≈ 𝐴) → ((𝐴 ∖ 𝑦) ∖ {𝑧}) ≈ 𝐴)
26	24, 25	sylancom 586	. . . . . 6 ⊢ ((ω ≼ 𝐴 ∧ (𝐴 ∖ 𝑦) ≈ 𝐴) → ((𝐴 ∖ 𝑦) ∖ {𝑧}) ≈ 𝐴)
27	26	ex 411	. . . . 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 9187	. 2 ⊢ (𝐵 ∈ Fin → (ω ≼ 𝐴 → (𝐴 ∖ 𝐵) ≈ 𝐴))
31	30	impcom 406	1 ⊢ ((ω ≼ 𝐴 ∧ 𝐵 ∈ Fin) → (𝐴 ∖ 𝐵) ≈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  Vcvv 3463   ∖ cdif 3942   ∪ cun 3943  ∅c0 4323  {csn 4629   class class class wbr 5148  ωcom 7869   ≈ cen 8959   ≼ cdom 8960  Fincfn 8962

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5299  ax-nul 5306  ax-pow 5364  ax-pr 5428  ax-un 7739

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3965  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  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 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6499  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-om 7870  df-er 8723  df-en 8963  df-dom 8964  df-fin 8966

This theorem is referenced by: (None)