Theorem difinfinf 7176

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 3276	. . 3 ⊢ (𝑤 = ∅ → (𝐴 ∖ 𝑤) = (𝐴 ∖ ∅))
2	1	breq2d 4046	. 2 ⊢ (𝑤 = ∅ → (ω ≼ (𝐴 ∖ 𝑤) ↔ ω ≼ (𝐴 ∖ ∅)))
3		difeq2 3276	. . 3 ⊢ (𝑤 = 𝑢 → (𝐴 ∖ 𝑤) = (𝐴 ∖ 𝑢))
4	3	breq2d 4046	. 2 ⊢ (𝑤 = 𝑢 → (ω ≼ (𝐴 ∖ 𝑤) ↔ ω ≼ (𝐴 ∖ 𝑢)))
5		difeq2 3276	. . 3 ⊢ (𝑤 = (𝑢 ∪ {𝑣}) → (𝐴 ∖ 𝑤) = (𝐴 ∖ (𝑢 ∪ {𝑣})))
6	5	breq2d 4046	. 2 ⊢ (𝑤 = (𝑢 ∪ {𝑣}) → (ω ≼ (𝐴 ∖ 𝑤) ↔ ω ≼ (𝐴 ∖ (𝑢 ∪ {𝑣}))))
7		difeq2 3276	. . 3 ⊢ (𝑤 = 𝐵 → (𝐴 ∖ 𝑤) = (𝐴 ∖ 𝐵))
8	7	breq2d 4046	. 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 3522	. . 3 ⊢ (𝐴 ∖ ∅) = 𝐴
11	9, 10	breqtrrdi 4076	. 2 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ∈ Fin)) → ω ≼ (𝐴 ∖ ∅))
12		difss 3290	. . . . . . 7 ⊢ (𝐴 ∖ 𝑢) ⊆ 𝐴
13		ssralv 3248	. . . . . . . . 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 2560	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ (𝐴 ∖ 𝑢)DECID 𝑥 = 𝑦)
16		ssralv 3248	. . . . . . 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 3299	. . . . . . 7 ⊢ (𝐵 ⊆ 𝐴 → (𝐵 ∖ 𝑢) ⊆ (𝐴 ∖ 𝑢))
24	23	sseld 3183	. . . . . 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 7175	. . . . 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 1249	. . . 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 3424	. . . 4 ⊢ (𝐴 ∖ (𝑢 ∪ {𝑣})) = ((𝐴 ∖ 𝑢) ∖ {𝑣})
29	27, 28	breqtrrdi 4076	. . 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 6962	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 835   = wceq 1364   ∈ wcel 2167  ∀wral 2475   ∖ cdif 3154   ∪ cun 3155   ⊆ wss 3157  ∅c0 3451  {csn 3623   class class class wbr 4034  ωcom 4627   ≼ cdom 6807  Fincfn 6808

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-1st 6207  df-2nd 6208  df-1o 6483  df-er 6601  df-en 6809  df-dom 6810  df-fin 6811  df-dju 7113  df-inl 7122  df-inr 7123  df-case 7159

This theorem is referenced by:  inffinp1  12671