Theorem infdiffi 9655

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 4116	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐴 ∖ 𝑥) = (𝐴 ∖ ∅))
2		dif0 4372	. . . . . 6 ⊢ (𝐴 ∖ ∅) = 𝐴
3	1, 2	eqtrdi 2788	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 ∖ 𝑥) = 𝐴)
4	3	breq1d 5158	. . . 4 ⊢ (𝑥 = ∅ → ((𝐴 ∖ 𝑥) ≈ 𝐴 ↔ 𝐴 ≈ 𝐴))
5	4	imbi2d 340	. . 3 ⊢ (𝑥 = ∅ → ((ω ≼ 𝐴 → (𝐴 ∖ 𝑥) ≈ 𝐴) ↔ (ω ≼ 𝐴 → 𝐴 ≈ 𝐴)))
6		difeq2 4116	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝑦))
7	6	breq1d 5158	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 ∖ 𝑥) ≈ 𝐴 ↔ (𝐴 ∖ 𝑦) ≈ 𝐴))
8	7	imbi2d 340	. . 3 ⊢ (𝑥 = 𝑦 → ((ω ≼ 𝐴 → (𝐴 ∖ 𝑥) ≈ 𝐴) ↔ (ω ≼ 𝐴 → (𝐴 ∖ 𝑦) ≈ 𝐴)))
9		difeq2 4116	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝐴 ∖ 𝑥) = (𝐴 ∖ (𝑦 ∪ {𝑧})))
10		difun1 4289	. . . . . 6 ⊢ (𝐴 ∖ (𝑦 ∪ {𝑧})) = ((𝐴 ∖ 𝑦) ∖ {𝑧})
11	9, 10	eqtrdi 2788	. . . . 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 340	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((ω ≼ 𝐴 → (𝐴 ∖ 𝑥) ≈ 𝐴) ↔ (ω ≼ 𝐴 → ((𝐴 ∖ 𝑦) ∖ {𝑧}) ≈ 𝐴)))
14		difeq2 4116	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝐵))
15	14	breq1d 5158	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝐴 ∖ 𝑥) ≈ 𝐴 ↔ (𝐴 ∖ 𝐵) ≈ 𝐴))
16	15	imbi2d 340	. . 3 ⊢ (𝑥 = 𝐵 → ((ω ≼ 𝐴 → (𝐴 ∖ 𝑥) ≈ 𝐴) ↔ (ω ≼ 𝐴 → (𝐴 ∖ 𝐵) ≈ 𝐴)))
17		reldom 8947	. . . . 5 ⊢ Rel ≼
18	17	brrelex2i 5733	. . . 4 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V)
19		enrefg 8982	. . . 4 ⊢ (𝐴 ∈ V → 𝐴 ≈ 𝐴)
20	18, 19	syl 17	. . 3 ⊢ (ω ≼ 𝐴 → 𝐴 ≈ 𝐴)
21		domen2 9122	. . . . . . . . 9 ⊢ ((𝐴 ∖ 𝑦) ≈ 𝐴 → (ω ≼ (𝐴 ∖ 𝑦) ↔ ω ≼ 𝐴))
22	21	biimparc 480	. . . . . . . 8 ⊢ ((ω ≼ 𝐴 ∧ (𝐴 ∖ 𝑦) ≈ 𝐴) → ω ≼ (𝐴 ∖ 𝑦))
23		infdifsn 9654	. . . . . . . 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 9004	. . . . . . 7 ⊢ ((((𝐴 ∖ 𝑦) ∖ {𝑧}) ≈ (𝐴 ∖ 𝑦) ∧ (𝐴 ∖ 𝑦) ≈ 𝐴) → ((𝐴 ∖ 𝑦) ∖ {𝑧}) ≈ 𝐴)
26	24, 25	sylancom 588	. . . . . 6 ⊢ ((ω ≼ 𝐴 ∧ (𝐴 ∖ 𝑦) ≈ 𝐴) → ((𝐴 ∖ 𝑦) ∖ {𝑧}) ≈ 𝐴)
27	26	ex 413	. . . . 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 9166	. 2 ⊢ (𝐵 ∈ Fin → (ω ≼ 𝐴 → (𝐴 ∖ 𝐵) ≈ 𝐴))
31	30	impcom 408	1 ⊢ ((ω ≼ 𝐴 ∧ 𝐵 ∈ Fin) → (𝐴 ∖ 𝐵) ≈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3474   ∖ cdif 3945   ∪ cun 3946  ∅c0 4322  {csn 4628   class class class wbr 5148  ωcom 7857   ≈ cen 8938   ≼ cdom 8939  Fincfn 8941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-om 7858  df-er 8705  df-en 8942  df-dom 8943  df-fin 8945

This theorem is referenced by: (None)