Theorem difinfinf 7300

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 3319	. . 3 ⊢ (𝑤 = ∅ → (𝐴 ∖ 𝑤) = (𝐴 ∖ ∅))
2	1	breq2d 4100	. 2 ⊢ (𝑤 = ∅ → (ω ≼ (𝐴 ∖ 𝑤) ↔ ω ≼ (𝐴 ∖ ∅)))
3		difeq2 3319	. . 3 ⊢ (𝑤 = 𝑢 → (𝐴 ∖ 𝑤) = (𝐴 ∖ 𝑢))
4	3	breq2d 4100	. 2 ⊢ (𝑤 = 𝑢 → (ω ≼ (𝐴 ∖ 𝑤) ↔ ω ≼ (𝐴 ∖ 𝑢)))
5		difeq2 3319	. . 3 ⊢ (𝑤 = (𝑢 ∪ {𝑣}) → (𝐴 ∖ 𝑤) = (𝐴 ∖ (𝑢 ∪ {𝑣})))
6	5	breq2d 4100	. 2 ⊢ (𝑤 = (𝑢 ∪ {𝑣}) → (ω ≼ (𝐴 ∖ 𝑤) ↔ ω ≼ (𝐴 ∖ (𝑢 ∪ {𝑣}))))
7		difeq2 3319	. . 3 ⊢ (𝑤 = 𝐵 → (𝐴 ∖ 𝑤) = (𝐴 ∖ 𝐵))
8	7	breq2d 4100	. 2 ⊢ (𝑤 = 𝐵 → (ω ≼ (𝐴 ∖ 𝑤) ↔ ω ≼ (𝐴 ∖ 𝐵)))
9		simplr 529	. . 3 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ∈ Fin)) → ω ≼ 𝐴)
10		dif0 3565	. . 3 ⊢ (𝐴 ∖ ∅) = 𝐴
11	9, 10	breqtrrdi 4130	. 2 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ∈ Fin)) → ω ≼ (𝐴 ∖ ∅))
12		difss 3333	. . . . . . 7 ⊢ (𝐴 ∖ 𝑢) ⊆ 𝐴
13		ssralv 3291	. . . . . . . . 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 2595	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ (𝐴 ∖ 𝑢)DECID 𝑥 = 𝑦)
16		ssralv 3291	. . . . . . 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 531	. . . . . . 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 538	. . . . . 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 3342	. . . . . . 7 ⊢ (𝐵 ⊆ 𝐴 → (𝐵 ∖ 𝑢) ⊆ (𝐴 ∖ 𝑢))
24	23	sseld 3226	. . . . . 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 7299	. . . . 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 1273	. . . 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 3467	. . . 4 ⊢ (𝐴 ∖ (𝑢 ∪ {𝑣})) = ((𝐴 ∖ 𝑢) ∖ {𝑣})
29	27, 28	breqtrrdi 4130	. . 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 533	. 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 7081	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 841   = wceq 1397   ∈ wcel 2202  ∀wral 2510   ∖ cdif 3197   ∪ cun 3198   ⊆ wss 3200  ∅c0 3494  {csn 3669   class class class wbr 4088  ωcom 4688   ≼ cdom 6908  Fincfn 6909

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-1st 6303  df-2nd 6304  df-1o 6582  df-er 6702  df-en 6910  df-dom 6911  df-fin 6912  df-dju 7237  df-inl 7246  df-inr 7247  df-case 7283

This theorem is referenced by:  inffinp1  13052