Theorem infdiffi 9105

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 4044	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐴 ∖ 𝑥) = (𝐴 ∖ ∅))
2		dif0 4286	. . . . . 6 ⊢ (𝐴 ∖ ∅) = 𝐴
3	1, 2	eqtrdi 2849	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 ∖ 𝑥) = 𝐴)
4	3	breq1d 5040	. . . 4 ⊢ (𝑥 = ∅ → ((𝐴 ∖ 𝑥) ≈ 𝐴 ↔ 𝐴 ≈ 𝐴))
5	4	imbi2d 344	. . 3 ⊢ (𝑥 = ∅ → ((ω ≼ 𝐴 → (𝐴 ∖ 𝑥) ≈ 𝐴) ↔ (ω ≼ 𝐴 → 𝐴 ≈ 𝐴)))
6		difeq2 4044	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝑦))
7	6	breq1d 5040	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 ∖ 𝑥) ≈ 𝐴 ↔ (𝐴 ∖ 𝑦) ≈ 𝐴))
8	7	imbi2d 344	. . 3 ⊢ (𝑥 = 𝑦 → ((ω ≼ 𝐴 → (𝐴 ∖ 𝑥) ≈ 𝐴) ↔ (ω ≼ 𝐴 → (𝐴 ∖ 𝑦) ≈ 𝐴)))
9		difeq2 4044	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝐴 ∖ 𝑥) = (𝐴 ∖ (𝑦 ∪ {𝑧})))
10		difun1 4214	. . . . . 6 ⊢ (𝐴 ∖ (𝑦 ∪ {𝑧})) = ((𝐴 ∖ 𝑦) ∖ {𝑧})
11	9, 10	eqtrdi 2849	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝐴 ∖ 𝑥) = ((𝐴 ∖ 𝑦) ∖ {𝑧}))
12	11	breq1d 5040	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐴 ∖ 𝑥) ≈ 𝐴 ↔ ((𝐴 ∖ 𝑦) ∖ {𝑧}) ≈ 𝐴))
13	12	imbi2d 344	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((ω ≼ 𝐴 → (𝐴 ∖ 𝑥) ≈ 𝐴) ↔ (ω ≼ 𝐴 → ((𝐴 ∖ 𝑦) ∖ {𝑧}) ≈ 𝐴)))
14		difeq2 4044	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝐵))
15	14	breq1d 5040	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝐴 ∖ 𝑥) ≈ 𝐴 ↔ (𝐴 ∖ 𝐵) ≈ 𝐴))
16	15	imbi2d 344	. . 3 ⊢ (𝑥 = 𝐵 → ((ω ≼ 𝐴 → (𝐴 ∖ 𝑥) ≈ 𝐴) ↔ (ω ≼ 𝐴 → (𝐴 ∖ 𝐵) ≈ 𝐴)))
17		reldom 8498	. . . . 5 ⊢ Rel ≼
18	17	brrelex2i 5573	. . . 4 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V)
19		enrefg 8524	. . . 4 ⊢ (𝐴 ∈ V → 𝐴 ≈ 𝐴)
20	18, 19	syl 17	. . 3 ⊢ (ω ≼ 𝐴 → 𝐴 ≈ 𝐴)
21		domen2 8644	. . . . . . . . 9 ⊢ ((𝐴 ∖ 𝑦) ≈ 𝐴 → (ω ≼ (𝐴 ∖ 𝑦) ↔ ω ≼ 𝐴))
22	21	biimparc 483	. . . . . . . 8 ⊢ ((ω ≼ 𝐴 ∧ (𝐴 ∖ 𝑦) ≈ 𝐴) → ω ≼ (𝐴 ∖ 𝑦))
23		infdifsn 9104	. . . . . . . 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 8544	. . . . . . 7 ⊢ ((((𝐴 ∖ 𝑦) ∖ {𝑧}) ≈ (𝐴 ∖ 𝑦) ∧ (𝐴 ∖ 𝑦) ≈ 𝐴) → ((𝐴 ∖ 𝑦) ∖ {𝑧}) ≈ 𝐴)
26	24, 25	sylancom 591	. . . . . 6 ⊢ ((ω ≼ 𝐴 ∧ (𝐴 ∖ 𝑦) ≈ 𝐴) → ((𝐴 ∖ 𝑦) ∖ {𝑧}) ≈ 𝐴)
27	26	ex 416	. . . . 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 8742	. 2 ⊢ (𝐵 ∈ Fin → (ω ≼ 𝐴 → (𝐴 ∖ 𝐵) ≈ 𝐴))
31	30	impcom 411	1 ⊢ ((ω ≼ 𝐴 ∧ 𝐵 ∈ Fin) → (𝐴 ∖ 𝐵) ≈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3441   ∖ cdif 3878   ∪ cun 3879  ∅c0 4243  {csn 4525   class class class wbr 5030  ωcom 7560   ≈ cen 8489   ≼ cdom 8490  Fincfn 8492

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-om 7561  df-1o 8085  df-er 8272  df-en 8493  df-dom 8494  df-fin 8496

This theorem is referenced by: (None)