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Theorem difinfinf 7130
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 3262 . . 3 (𝑤 = ∅ → (𝐴𝑤) = (𝐴 ∖ ∅))
21breq2d 4030 . 2 (𝑤 = ∅ → (ω ≼ (𝐴𝑤) ↔ ω ≼ (𝐴 ∖ ∅)))
3 difeq2 3262 . . 3 (𝑤 = 𝑢 → (𝐴𝑤) = (𝐴𝑢))
43breq2d 4030 . 2 (𝑤 = 𝑢 → (ω ≼ (𝐴𝑤) ↔ ω ≼ (𝐴𝑢)))
5 difeq2 3262 . . 3 (𝑤 = (𝑢 ∪ {𝑣}) → (𝐴𝑤) = (𝐴 ∖ (𝑢 ∪ {𝑣})))
65breq2d 4030 . 2 (𝑤 = (𝑢 ∪ {𝑣}) → (ω ≼ (𝐴𝑤) ↔ ω ≼ (𝐴 ∖ (𝑢 ∪ {𝑣}))))
7 difeq2 3262 . . 3 (𝑤 = 𝐵 → (𝐴𝑤) = (𝐴𝐵))
87breq2d 4030 . 2 (𝑤 = 𝐵 → (ω ≼ (𝐴𝑤) ↔ ω ≼ (𝐴𝐵)))
9 simplr 528 . . 3 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵𝐴𝐵 ∈ Fin)) → ω ≼ 𝐴)
10 dif0 3508 . . 3 (𝐴 ∖ ∅) = 𝐴
119, 10breqtrrdi 4060 . 2 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵𝐴𝐵 ∈ Fin)) → ω ≼ (𝐴 ∖ ∅))
12 difss 3276 . . . . . . 7 (𝐴𝑢) ⊆ 𝐴
13 ssralv 3234 . . . . . . . . 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 2553 . . . . . . 7 (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 → ∀𝑥𝐴𝑦 ∈ (𝐴𝑢)DECID 𝑥 = 𝑦)
16 ssralv 3234 . . . . . . 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 496 . . . . 5 ((((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵𝐴𝐵 ∈ Fin)) ∧ 𝑢 ∈ Fin) ∧ (𝑢𝐵𝑣 ∈ (𝐵𝑢))) ∧ ω ≼ (𝐴𝑢)) → ∀𝑥 ∈ (𝐴𝑢)∀𝑦 ∈ (𝐴𝑢)DECID 𝑥 = 𝑦)
19 simpr 110 . . . . 5 ((((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵𝐴𝐵 ∈ Fin)) ∧ 𝑢 ∈ Fin) ∧ (𝑢𝐵𝑣 ∈ (𝐵𝑢))) ∧ ω ≼ (𝐴𝑢)) → ω ≼ (𝐴𝑢))
20 simprl 529 . . . . . . 7 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵𝐴𝐵 ∈ Fin)) → 𝐵𝐴)
2120ad3antrrr 492 . . . . . 6 ((((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵𝐴𝐵 ∈ Fin)) ∧ 𝑢 ∈ Fin) ∧ (𝑢𝐵𝑣 ∈ (𝐵𝑢))) ∧ ω ≼ (𝐴𝑢)) → 𝐵𝐴)
22 simplrr 536 . . . . . 6 ((((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵𝐴𝐵 ∈ Fin)) ∧ 𝑢 ∈ Fin) ∧ (𝑢𝐵𝑣 ∈ (𝐵𝑢))) ∧ ω ≼ (𝐴𝑢)) → 𝑣 ∈ (𝐵𝑢))
23 ssdif 3285 . . . . . . 7 (𝐵𝐴 → (𝐵𝑢) ⊆ (𝐴𝑢))
2423sseld 3169 . . . . . 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 7129 . . . . 5 ((∀𝑥 ∈ (𝐴𝑢)∀𝑦 ∈ (𝐴𝑢)DECID 𝑥 = 𝑦 ∧ ω ≼ (𝐴𝑢) ∧ 𝑣 ∈ (𝐴𝑢)) → ω ≼ ((𝐴𝑢) ∖ {𝑣}))
2718, 19, 25, 26syl3anc 1249 . . . 4 ((((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵𝐴𝐵 ∈ Fin)) ∧ 𝑢 ∈ Fin) ∧ (𝑢𝐵𝑣 ∈ (𝐵𝑢))) ∧ ω ≼ (𝐴𝑢)) → ω ≼ ((𝐴𝑢) ∖ {𝑣}))
28 difun1 3410 . . . 4 (𝐴 ∖ (𝑢 ∪ {𝑣})) = ((𝐴𝑢) ∖ {𝑣})
2927, 28breqtrrdi 4060 . . 3 ((((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵𝐴𝐵 ∈ Fin)) ∧ 𝑢 ∈ Fin) ∧ (𝑢𝐵𝑣 ∈ (𝐵𝑢))) ∧ ω ≼ (𝐴𝑢)) → ω ≼ (𝐴 ∖ (𝑢 ∪ {𝑣})))
3029ex 115 . 2 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵𝐴𝐵 ∈ Fin)) ∧ 𝑢 ∈ Fin) ∧ (𝑢𝐵𝑣 ∈ (𝐵𝑢))) → (ω ≼ (𝐴𝑢) → ω ≼ (𝐴 ∖ (𝑢 ∪ {𝑣}))))
31 simprr 531 . 2 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵𝐴𝐵 ∈ Fin)) → 𝐵 ∈ Fin)
322, 4, 6, 8, 11, 30, 31findcard2sd 6920 1 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵𝐴𝐵 ∈ Fin)) → ω ≼ (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  DECID wdc 835   = wceq 1364  wcel 2160  wral 2468  cdif 3141  cun 3142  wss 3144  c0 3437  {csn 3607   class class class wbr 4018  ωcom 4607  cdom 6765  Fincfn 6766
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
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 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-1st 6165  df-2nd 6166  df-1o 6441  df-er 6559  df-en 6767  df-dom 6768  df-fin 6769  df-dju 7067  df-inl 7076  df-inr 7077  df-case 7113
This theorem is referenced by:  inffinp1  12480
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