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Theorem difinfinf 6938
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 3154 . . 3 (𝑤 = ∅ → (𝐴𝑤) = (𝐴 ∖ ∅))
21breq2d 3907 . 2 (𝑤 = ∅ → (ω ≼ (𝐴𝑤) ↔ ω ≼ (𝐴 ∖ ∅)))
3 difeq2 3154 . . 3 (𝑤 = 𝑢 → (𝐴𝑤) = (𝐴𝑢))
43breq2d 3907 . 2 (𝑤 = 𝑢 → (ω ≼ (𝐴𝑤) ↔ ω ≼ (𝐴𝑢)))
5 difeq2 3154 . . 3 (𝑤 = (𝑢 ∪ {𝑣}) → (𝐴𝑤) = (𝐴 ∖ (𝑢 ∪ {𝑣})))
65breq2d 3907 . 2 (𝑤 = (𝑢 ∪ {𝑣}) → (ω ≼ (𝐴𝑤) ↔ ω ≼ (𝐴 ∖ (𝑢 ∪ {𝑣}))))
7 difeq2 3154 . . 3 (𝑤 = 𝐵 → (𝐴𝑤) = (𝐴𝐵))
87breq2d 3907 . 2 (𝑤 = 𝐵 → (ω ≼ (𝐴𝑤) ↔ ω ≼ (𝐴𝐵)))
9 simplr 502 . . 3 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵𝐴𝐵 ∈ Fin)) → ω ≼ 𝐴)
10 dif0 3399 . . 3 (𝐴 ∖ ∅) = 𝐴
119, 10syl6breqr 3935 . 2 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵𝐴𝐵 ∈ Fin)) → ω ≼ (𝐴 ∖ ∅))
12 difss 3168 . . . . . . 7 (𝐴𝑢) ⊆ 𝐴
13 ssralv 3127 . . . . . . . . 9 ((𝐴𝑢) ⊆ 𝐴 → (∀𝑦𝐴 DECID 𝑥 = 𝑦 → ∀𝑦 ∈ (𝐴𝑢)DECID 𝑥 = 𝑦))
1412, 13ax-mp 7 . . . . . . . 8 (∀𝑦𝐴 DECID 𝑥 = 𝑦 → ∀𝑦 ∈ (𝐴𝑢)DECID 𝑥 = 𝑦)
1514ralimi 2469 . . . . . . 7 (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 → ∀𝑥𝐴𝑦 ∈ (𝐴𝑢)DECID 𝑥 = 𝑦)
16 ssralv 3127 . . . . . . 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 485 . . . . 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 503 . . . . . . 7 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵𝐴𝐵 ∈ Fin)) → 𝐵𝐴)
2120ad3antrrr 481 . . . . . 6 ((((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵𝐴𝐵 ∈ Fin)) ∧ 𝑢 ∈ Fin) ∧ (𝑢𝐵𝑣 ∈ (𝐵𝑢))) ∧ ω ≼ (𝐴𝑢)) → 𝐵𝐴)
22 simplrr 508 . . . . . 6 ((((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵𝐴𝐵 ∈ Fin)) ∧ 𝑢 ∈ Fin) ∧ (𝑢𝐵𝑣 ∈ (𝐵𝑢))) ∧ ω ≼ (𝐴𝑢)) → 𝑣 ∈ (𝐵𝑢))
23 ssdif 3177 . . . . . . 7 (𝐵𝐴 → (𝐵𝑢) ⊆ (𝐴𝑢))
2423sseld 3062 . . . . . 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 6937 . . . . 5 ((∀𝑥 ∈ (𝐴𝑢)∀𝑦 ∈ (𝐴𝑢)DECID 𝑥 = 𝑦 ∧ ω ≼ (𝐴𝑢) ∧ 𝑣 ∈ (𝐴𝑢)) → ω ≼ ((𝐴𝑢) ∖ {𝑣}))
2718, 19, 25, 26syl3anc 1199 . . . 4 ((((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵𝐴𝐵 ∈ Fin)) ∧ 𝑢 ∈ Fin) ∧ (𝑢𝐵𝑣 ∈ (𝐵𝑢))) ∧ ω ≼ (𝐴𝑢)) → ω ≼ ((𝐴𝑢) ∖ {𝑣}))
28 difun1 3302 . . . 4 (𝐴 ∖ (𝑢 ∪ {𝑣})) = ((𝐴𝑢) ∖ {𝑣})
2927, 28syl6breqr 3935 . . 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 504 . 2 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵𝐴𝐵 ∈ Fin)) → 𝐵 ∈ Fin)
322, 4, 6, 8, 11, 30, 31findcard2sd 6739 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 802   = wceq 1314  wcel 1463  wral 2390  cdif 3034  cun 3035  wss 3037  c0 3329  {csn 3493   class class class wbr 3895  ωcom 4464  cdom 6587  Fincfn 6588
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-iinf 4462
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-if 3441  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-id 4175  df-iord 4248  df-on 4250  df-suc 4253  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-1st 5992  df-2nd 5993  df-1o 6267  df-er 6383  df-en 6589  df-dom 6590  df-fin 6591  df-dju 6875  df-inl 6884  df-inr 6885  df-case 6921
This theorem is referenced by:  inffinp1  11784
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