Theorem difinfinf 6986

Description: An infinite set minus a finite subset is infinite. We require that the set has decidable equality. (Contributed by Jim Kingdon, 8-Aug-2023.)

Assertion

Ref	Expression
difinfinf	⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ∈ Fin)) → ω ≼ (𝐴 ∖ 𝐵))

Distinct variable group:   𝑥,𝐴,𝑦

Allowed substitution hints:   𝐵(𝑥,𝑦)

Proof of Theorem difinfinf

Dummy variables 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		difeq2 3188	. . 3 ⊢ (𝑤 = ∅ → (𝐴 ∖ 𝑤) = (𝐴 ∖ ∅))
2	1	breq2d 3941	. 2 ⊢ (𝑤 = ∅ → (ω ≼ (𝐴 ∖ 𝑤) ↔ ω ≼ (𝐴 ∖ ∅)))
3		difeq2 3188	. . 3 ⊢ (𝑤 = 𝑢 → (𝐴 ∖ 𝑤) = (𝐴 ∖ 𝑢))
4	3	breq2d 3941	. 2 ⊢ (𝑤 = 𝑢 → (ω ≼ (𝐴 ∖ 𝑤) ↔ ω ≼ (𝐴 ∖ 𝑢)))
5		difeq2 3188	. . 3 ⊢ (𝑤 = (𝑢 ∪ {𝑣}) → (𝐴 ∖ 𝑤) = (𝐴 ∖ (𝑢 ∪ {𝑣})))
6	5	breq2d 3941	. 2 ⊢ (𝑤 = (𝑢 ∪ {𝑣}) → (ω ≼ (𝐴 ∖ 𝑤) ↔ ω ≼ (𝐴 ∖ (𝑢 ∪ {𝑣}))))
7		difeq2 3188	. . 3 ⊢ (𝑤 = 𝐵 → (𝐴 ∖ 𝑤) = (𝐴 ∖ 𝐵))
8	7	breq2d 3941	. 2 ⊢ (𝑤 = 𝐵 → (ω ≼ (𝐴 ∖ 𝑤) ↔ ω ≼ (𝐴 ∖ 𝐵)))
9		simplr 519	. . 3 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ∈ Fin)) → ω ≼ 𝐴)
10		dif0 3433	. . 3 ⊢ (𝐴 ∖ ∅) = 𝐴
11	9, 10	breqtrrdi 3970	. 2 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ∈ Fin)) → ω ≼ (𝐴 ∖ ∅))
12		difss 3202	. . . . . . 7 ⊢ (𝐴 ∖ 𝑢) ⊆ 𝐴
13		ssralv 3161	. . . . . . . . 9 ⊢ ((𝐴 ∖ 𝑢) ⊆ 𝐴 → (∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 → ∀𝑦 ∈ (𝐴 ∖ 𝑢)DECID 𝑥 = 𝑦))
14	12, 13	ax-mp 5	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 → ∀𝑦 ∈ (𝐴 ∖ 𝑢)DECID 𝑥 = 𝑦)
15	14	ralimi 2495	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ (𝐴 ∖ 𝑢)DECID 𝑥 = 𝑦)
16		ssralv 3161	. . . . . . 7 ⊢ ((𝐴 ∖ 𝑢) ⊆ 𝐴 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ (𝐴 ∖ 𝑢)DECID 𝑥 = 𝑦 → ∀𝑥 ∈ (𝐴 ∖ 𝑢)∀𝑦 ∈ (𝐴 ∖ 𝑢)DECID 𝑥 = 𝑦))
17	12, 15, 16	mpsyl 65	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 → ∀𝑥 ∈ (𝐴 ∖ 𝑢)∀𝑦 ∈ (𝐴 ∖ 𝑢)DECID 𝑥 = 𝑦)
18	17	ad5antr 487	. . . . 5 ⊢ ((((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ∈ Fin)) ∧ 𝑢 ∈ Fin) ∧ (𝑢 ⊆ 𝐵 ∧ 𝑣 ∈ (𝐵 ∖ 𝑢))) ∧ ω ≼ (𝐴 ∖ 𝑢)) → ∀𝑥 ∈ (𝐴 ∖ 𝑢)∀𝑦 ∈ (𝐴 ∖ 𝑢)DECID 𝑥 = 𝑦)
19		simpr 109	. . . . 5 ⊢ ((((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ∈ Fin)) ∧ 𝑢 ∈ Fin) ∧ (𝑢 ⊆ 𝐵 ∧ 𝑣 ∈ (𝐵 ∖ 𝑢))) ∧ ω ≼ (𝐴 ∖ 𝑢)) → ω ≼ (𝐴 ∖ 𝑢))
20		simprl 520	. . . . . . 7 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ∈ Fin)) → 𝐵 ⊆ 𝐴)
21	20	ad3antrrr 483	. . . . . 6 ⊢ ((((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ∈ Fin)) ∧ 𝑢 ∈ Fin) ∧ (𝑢 ⊆ 𝐵 ∧ 𝑣 ∈ (𝐵 ∖ 𝑢))) ∧ ω ≼ (𝐴 ∖ 𝑢)) → 𝐵 ⊆ 𝐴)
22		simplrr 525	. . . . . 6 ⊢ ((((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ∈ Fin)) ∧ 𝑢 ∈ Fin) ∧ (𝑢 ⊆ 𝐵 ∧ 𝑣 ∈ (𝐵 ∖ 𝑢))) ∧ ω ≼ (𝐴 ∖ 𝑢)) → 𝑣 ∈ (𝐵 ∖ 𝑢))
23		ssdif 3211	. . . . . . 7 ⊢ (𝐵 ⊆ 𝐴 → (𝐵 ∖ 𝑢) ⊆ (𝐴 ∖ 𝑢))
24	23	sseld 3096	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 → (𝑣 ∈ (𝐵 ∖ 𝑢) → 𝑣 ∈ (𝐴 ∖ 𝑢)))
25	21, 22, 24	sylc 62	. . . . 5 ⊢ ((((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ∈ Fin)) ∧ 𝑢 ∈ Fin) ∧ (𝑢 ⊆ 𝐵 ∧ 𝑣 ∈ (𝐵 ∖ 𝑢))) ∧ ω ≼ (𝐴 ∖ 𝑢)) → 𝑣 ∈ (𝐴 ∖ 𝑢))
26		difinfsn 6985	. . . . 5 ⊢ ((∀𝑥 ∈ (𝐴 ∖ 𝑢)∀𝑦 ∈ (𝐴 ∖ 𝑢)DECID 𝑥 = 𝑦 ∧ ω ≼ (𝐴 ∖ 𝑢) ∧ 𝑣 ∈ (𝐴 ∖ 𝑢)) → ω ≼ ((𝐴 ∖ 𝑢) ∖ {𝑣}))
27	18, 19, 25, 26	syl3anc 1216	. . . 4 ⊢ ((((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ∈ Fin)) ∧ 𝑢 ∈ Fin) ∧ (𝑢 ⊆ 𝐵 ∧ 𝑣 ∈ (𝐵 ∖ 𝑢))) ∧ ω ≼ (𝐴 ∖ 𝑢)) → ω ≼ ((𝐴 ∖ 𝑢) ∖ {𝑣}))
28		difun1 3336	. . . 4 ⊢ (𝐴 ∖ (𝑢 ∪ {𝑣})) = ((𝐴 ∖ 𝑢) ∖ {𝑣})
29	27, 28	breqtrrdi 3970	. . 3 ⊢ ((((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ∈ Fin)) ∧ 𝑢 ∈ Fin) ∧ (𝑢 ⊆ 𝐵 ∧ 𝑣 ∈ (𝐵 ∖ 𝑢))) ∧ ω ≼ (𝐴 ∖ 𝑢)) → ω ≼ (𝐴 ∖ (𝑢 ∪ {𝑣})))
30	29	ex 114	. 2 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ∈ Fin)) ∧ 𝑢 ∈ Fin) ∧ (𝑢 ⊆ 𝐵 ∧ 𝑣 ∈ (𝐵 ∖ 𝑢))) → (ω ≼ (𝐴 ∖ 𝑢) → ω ≼ (𝐴 ∖ (𝑢 ∪ {𝑣}))))
31		simprr 521	. 2 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ∈ Fin)) → 𝐵 ∈ Fin)
32	2, 4, 6, 8, 11, 30, 31	findcard2sd 6786	1 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ∈ Fin)) → ω ≼ (𝐴 ∖ 𝐵))

Syntax hints:   → wi 4   ∧ wa 103  DECID wdc 819   = wceq 1331   ∈ wcel 1480  ∀wral 2416   ∖ cdif 3068   ∪ cun 3069   ⊆ wss 3071  ∅c0 3363  {csn 3527   class class class wbr 3929  ωcom 4504   ≼ cdom 6633  Fincfn 6634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-1st 6038  df-2nd 6039  df-1o 6313  df-er 6429  df-en 6635  df-dom 6636  df-fin 6637  df-dju 6923  df-inl 6932  df-inr 6933  df-case 6969

This theorem is referenced by:  inffinp1  11942