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Theorem difinfinf 7074
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 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵𝐴𝐵 ∈ Fin)) → ω ≼ (𝐴𝐵))
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.
StepHypRef Expression
1 difeq2 3239 . . 3 (𝑤 = ∅ → (𝐴𝑤) = (𝐴 ∖ ∅))
21breq2d 3999 . 2 (𝑤 = ∅ → (ω ≼ (𝐴𝑤) ↔ ω ≼ (𝐴 ∖ ∅)))
3 difeq2 3239 . . 3 (𝑤 = 𝑢 → (𝐴𝑤) = (𝐴𝑢))
43breq2d 3999 . 2 (𝑤 = 𝑢 → (ω ≼ (𝐴𝑤) ↔ ω ≼ (𝐴𝑢)))
5 difeq2 3239 . . 3 (𝑤 = (𝑢 ∪ {𝑣}) → (𝐴𝑤) = (𝐴 ∖ (𝑢 ∪ {𝑣})))
65breq2d 3999 . 2 (𝑤 = (𝑢 ∪ {𝑣}) → (ω ≼ (𝐴𝑤) ↔ ω ≼ (𝐴 ∖ (𝑢 ∪ {𝑣}))))
7 difeq2 3239 . . 3 (𝑤 = 𝐵 → (𝐴𝑤) = (𝐴𝐵))
87breq2d 3999 . 2 (𝑤 = 𝐵 → (ω ≼ (𝐴𝑤) ↔ ω ≼ (𝐴𝐵)))
9 simplr 525 . . 3 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵𝐴𝐵 ∈ Fin)) → ω ≼ 𝐴)
10 dif0 3484 . . 3 (𝐴 ∖ ∅) = 𝐴
119, 10breqtrrdi 4029 . 2 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵𝐴𝐵 ∈ Fin)) → ω ≼ (𝐴 ∖ ∅))
12 difss 3253 . . . . . . 7 (𝐴𝑢) ⊆ 𝐴
13 ssralv 3211 . . . . . . . . 9 ((𝐴𝑢) ⊆ 𝐴 → (∀𝑦𝐴 DECID 𝑥 = 𝑦 → ∀𝑦 ∈ (𝐴𝑢)DECID 𝑥 = 𝑦))
1412, 13ax-mp 5 . . . . . . . 8 (∀𝑦𝐴 DECID 𝑥 = 𝑦 → ∀𝑦 ∈ (𝐴𝑢)DECID 𝑥 = 𝑦)
1514ralimi 2533 . . . . . . 7 (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 → ∀𝑥𝐴𝑦 ∈ (𝐴𝑢)DECID 𝑥 = 𝑦)
16 ssralv 3211 . . . . . . 7 ((𝐴𝑢) ⊆ 𝐴 → (∀𝑥𝐴𝑦 ∈ (𝐴𝑢)DECID 𝑥 = 𝑦 → ∀𝑥 ∈ (𝐴𝑢)∀𝑦 ∈ (𝐴𝑢)DECID 𝑥 = 𝑦))
1712, 15, 16mpsyl 65 . . . . . 6 (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 → ∀𝑥 ∈ (𝐴𝑢)∀𝑦 ∈ (𝐴𝑢)DECID 𝑥 = 𝑦)
1817ad5antr 493 . . . . 5 ((((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵𝐴𝐵 ∈ Fin)) ∧ 𝑢 ∈ Fin) ∧ (𝑢𝐵𝑣 ∈ (𝐵𝑢))) ∧ ω ≼ (𝐴𝑢)) → ∀𝑥 ∈ (𝐴𝑢)∀𝑦 ∈ (𝐴𝑢)DECID 𝑥 = 𝑦)
19 simpr 109 . . . . 5 ((((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵𝐴𝐵 ∈ Fin)) ∧ 𝑢 ∈ Fin) ∧ (𝑢𝐵𝑣 ∈ (𝐵𝑢))) ∧ ω ≼ (𝐴𝑢)) → ω ≼ (𝐴𝑢))
20 simprl 526 . . . . . . 7 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵𝐴𝐵 ∈ Fin)) → 𝐵𝐴)
2120ad3antrrr 489 . . . . . 6 ((((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵𝐴𝐵 ∈ Fin)) ∧ 𝑢 ∈ Fin) ∧ (𝑢𝐵𝑣 ∈ (𝐵𝑢))) ∧ ω ≼ (𝐴𝑢)) → 𝐵𝐴)
22 simplrr 531 . . . . . 6 ((((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵𝐴𝐵 ∈ Fin)) ∧ 𝑢 ∈ Fin) ∧ (𝑢𝐵𝑣 ∈ (𝐵𝑢))) ∧ ω ≼ (𝐴𝑢)) → 𝑣 ∈ (𝐵𝑢))
23 ssdif 3262 . . . . . . 7 (𝐵𝐴 → (𝐵𝑢) ⊆ (𝐴𝑢))
2423sseld 3146 . . . . . 6 (𝐵𝐴 → (𝑣 ∈ (𝐵𝑢) → 𝑣 ∈ (𝐴𝑢)))
2521, 22, 24sylc 62 . . . . 5 ((((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵𝐴𝐵 ∈ Fin)) ∧ 𝑢 ∈ Fin) ∧ (𝑢𝐵𝑣 ∈ (𝐵𝑢))) ∧ ω ≼ (𝐴𝑢)) → 𝑣 ∈ (𝐴𝑢))
26 difinfsn 7073 . . . . 5 ((∀𝑥 ∈ (𝐴𝑢)∀𝑦 ∈ (𝐴𝑢)DECID 𝑥 = 𝑦 ∧ ω ≼ (𝐴𝑢) ∧ 𝑣 ∈ (𝐴𝑢)) → ω ≼ ((𝐴𝑢) ∖ {𝑣}))
2718, 19, 25, 26syl3anc 1233 . . . 4 ((((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵𝐴𝐵 ∈ Fin)) ∧ 𝑢 ∈ Fin) ∧ (𝑢𝐵𝑣 ∈ (𝐵𝑢))) ∧ ω ≼ (𝐴𝑢)) → ω ≼ ((𝐴𝑢) ∖ {𝑣}))
28 difun1 3387 . . . 4 (𝐴 ∖ (𝑢 ∪ {𝑣})) = ((𝐴𝑢) ∖ {𝑣})
2927, 28breqtrrdi 4029 . . 3 ((((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵𝐴𝐵 ∈ Fin)) ∧ 𝑢 ∈ Fin) ∧ (𝑢𝐵𝑣 ∈ (𝐵𝑢))) ∧ ω ≼ (𝐴𝑢)) → ω ≼ (𝐴 ∖ (𝑢 ∪ {𝑣})))
3029ex 114 . 2 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵𝐴𝐵 ∈ Fin)) ∧ 𝑢 ∈ Fin) ∧ (𝑢𝐵𝑣 ∈ (𝐵𝑢))) → (ω ≼ (𝐴𝑢) → ω ≼ (𝐴 ∖ (𝑢 ∪ {𝑣}))))
31 simprr 527 . 2 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵𝐴𝐵 ∈ Fin)) → 𝐵 ∈ Fin)
322, 4, 6, 8, 11, 30, 31findcard2sd 6866 1 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵𝐴𝐵 ∈ Fin)) → ω ≼ (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  DECID wdc 829   = wceq 1348  wcel 2141  wral 2448  cdif 3118  cun 3119  wss 3121  c0 3414  {csn 3581   class class class wbr 3987  ωcom 4572  cdom 6713  Fincfn 6714
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 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570
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 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-iord 4349  df-on 4351  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-1st 6116  df-2nd 6117  df-1o 6392  df-er 6509  df-en 6715  df-dom 6716  df-fin 6717  df-dju 7011  df-inl 7020  df-inr 7021  df-case 7057
This theorem is referenced by:  inffinp1  12371
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