Theorem difinfinf 7095

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 3247	. . 3 ⊢ (𝑤 = ∅ → (𝐴 ∖ 𝑤) = (𝐴 ∖ ∅))
2	1	breq2d 4013	. 2 ⊢ (𝑤 = ∅ → (ω ≼ (𝐴 ∖ 𝑤) ↔ ω ≼ (𝐴 ∖ ∅)))
3		difeq2 3247	. . 3 ⊢ (𝑤 = 𝑢 → (𝐴 ∖ 𝑤) = (𝐴 ∖ 𝑢))
4	3	breq2d 4013	. 2 ⊢ (𝑤 = 𝑢 → (ω ≼ (𝐴 ∖ 𝑤) ↔ ω ≼ (𝐴 ∖ 𝑢)))
5		difeq2 3247	. . 3 ⊢ (𝑤 = (𝑢 ∪ {𝑣}) → (𝐴 ∖ 𝑤) = (𝐴 ∖ (𝑢 ∪ {𝑣})))
6	5	breq2d 4013	. 2 ⊢ (𝑤 = (𝑢 ∪ {𝑣}) → (ω ≼ (𝐴 ∖ 𝑤) ↔ ω ≼ (𝐴 ∖ (𝑢 ∪ {𝑣}))))
7		difeq2 3247	. . 3 ⊢ (𝑤 = 𝐵 → (𝐴 ∖ 𝑤) = (𝐴 ∖ 𝐵))
8	7	breq2d 4013	. 2 ⊢ (𝑤 = 𝐵 → (ω ≼ (𝐴 ∖ 𝑤) ↔ ω ≼ (𝐴 ∖ 𝐵)))
9		simplr 528	. . 3 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ∈ Fin)) → ω ≼ 𝐴)
10		dif0 3493	. . 3 ⊢ (𝐴 ∖ ∅) = 𝐴
11	9, 10	breqtrrdi 4043	. 2 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ∈ Fin)) → ω ≼ (𝐴 ∖ ∅))
12		difss 3261	. . . . . . 7 ⊢ (𝐴 ∖ 𝑢) ⊆ 𝐴
13		ssralv 3219	. . . . . . . . 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 2540	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ (𝐴 ∖ 𝑢)DECID 𝑥 = 𝑦)
16		ssralv 3219	. . . . . . 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 496	. . . . 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 110	. . . . 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 529	. . . . . . 7 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ∈ Fin)) → 𝐵 ⊆ 𝐴)
21	20	ad3antrrr 492	. . . . . 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 536	. . . . . 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 3270	. . . . . . 7 ⊢ (𝐵 ⊆ 𝐴 → (𝐵 ∖ 𝑢) ⊆ (𝐴 ∖ 𝑢))
24	23	sseld 3154	. . . . . 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 7094	. . . . 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 1238	. . . 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 3395	. . . 4 ⊢ (𝐴 ∖ (𝑢 ∪ {𝑣})) = ((𝐴 ∖ 𝑢) ∖ {𝑣})
29	27, 28	breqtrrdi 4043	. . 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 115	. 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 531	. 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 6887	1 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ∈ Fin)) → ω ≼ (𝐴 ∖ 𝐵))

Syntax hints:   → wi 4   ∧ wa 104  DECID wdc 834   = wceq 1353   ∈ wcel 2148  ∀wral 2455   ∖ cdif 3126   ∪ cun 3127   ⊆ wss 3129  ∅c0 3422  {csn 3592   class class class wbr 4001  ωcom 4587   ≼ cdom 6734  Fincfn 6735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4116  ax-sep 4119  ax-nul 4127  ax-pow 4172  ax-pr 4207  ax-un 4431  ax-setind 4534  ax-iinf 4585

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3809  df-int 3844  df-iun 3887  df-br 4002  df-opab 4063  df-mpt 4064  df-tr 4100  df-id 4291  df-iord 4364  df-on 4366  df-suc 4369  df-iom 4588  df-xp 4630  df-rel 4631  df-cnv 4632  df-co 4633  df-dm 4634  df-rn 4635  df-res 4636  df-ima 4637  df-iota 5175  df-fun 5215  df-fn 5216  df-f 5217  df-f1 5218  df-fo 5219  df-f1o 5220  df-fv 5221  df-1st 6136  df-2nd 6137  df-1o 6412  df-er 6530  df-en 6736  df-dom 6737  df-fin 6738  df-dju 7032  df-inl 7041  df-inr 7042  df-case 7078

This theorem is referenced by:  inffinp1  12420