Theorem difinfinf 7078

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 3239	. . 3 ⊢ (𝑤 = ∅ → (𝐴 ∖ 𝑤) = (𝐴 ∖ ∅))
2	1	breq2d 4001	. 2 ⊢ (𝑤 = ∅ → (ω ≼ (𝐴 ∖ 𝑤) ↔ ω ≼ (𝐴 ∖ ∅)))
3		difeq2 3239	. . 3 ⊢ (𝑤 = 𝑢 → (𝐴 ∖ 𝑤) = (𝐴 ∖ 𝑢))
4	3	breq2d 4001	. 2 ⊢ (𝑤 = 𝑢 → (ω ≼ (𝐴 ∖ 𝑤) ↔ ω ≼ (𝐴 ∖ 𝑢)))
5		difeq2 3239	. . 3 ⊢ (𝑤 = (𝑢 ∪ {𝑣}) → (𝐴 ∖ 𝑤) = (𝐴 ∖ (𝑢 ∪ {𝑣})))
6	5	breq2d 4001	. 2 ⊢ (𝑤 = (𝑢 ∪ {𝑣}) → (ω ≼ (𝐴 ∖ 𝑤) ↔ ω ≼ (𝐴 ∖ (𝑢 ∪ {𝑣}))))
7		difeq2 3239	. . 3 ⊢ (𝑤 = 𝐵 → (𝐴 ∖ 𝑤) = (𝐴 ∖ 𝐵))
8	7	breq2d 4001	. 2 ⊢ (𝑤 = 𝐵 → (ω ≼ (𝐴 ∖ 𝑤) ↔ ω ≼ (𝐴 ∖ 𝐵)))
9		simplr 525	. . 3 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ∈ Fin)) → ω ≼ 𝐴)
10		dif0 3485	. . 3 ⊢ (𝐴 ∖ ∅) = 𝐴
11	9, 10	breqtrrdi 4031	. 2 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ∈ Fin)) → ω ≼ (𝐴 ∖ ∅))
12		difss 3253	. . . . . . 7 ⊢ (𝐴 ∖ 𝑢) ⊆ 𝐴
13		ssralv 3211	. . . . . . . . 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 2533	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ (𝐴 ∖ 𝑢)DECID 𝑥 = 𝑦)
16		ssralv 3211	. . . . . . 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 493	. . . . 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 526	. . . . . . 7 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ∈ Fin)) → 𝐵 ⊆ 𝐴)
21	20	ad3antrrr 489	. . . . . 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 531	. . . . . 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 3262	. . . . . . 7 ⊢ (𝐵 ⊆ 𝐴 → (𝐵 ∖ 𝑢) ⊆ (𝐴 ∖ 𝑢))
24	23	sseld 3146	. . . . . 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 7077	. . . . 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 1233	. . . 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 3387	. . . 4 ⊢ (𝐴 ∖ (𝑢 ∪ {𝑣})) = ((𝐴 ∖ 𝑢) ∖ {𝑣})
29	27, 28	breqtrrdi 4031	. . 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 527	. 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 6870	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 829   = wceq 1348   ∈ wcel 2141  ∀wral 2448   ∖ cdif 3118   ∪ cun 3119   ⊆ wss 3121  ∅c0 3414  {csn 3583   class class class wbr 3989  ωcom 4574   ≼ cdom 6717  Fincfn 6718

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-1st 6119  df-2nd 6120  df-1o 6395  df-er 6513  df-en 6719  df-dom 6720  df-fin 6721  df-dju 7015  df-inl 7024  df-inr 7025  df-case 7061

This theorem is referenced by:  inffinp1  12384