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Mirrors > Home > ILE Home > Th. List > inffinp1 | GIF version |
Description: An infinite set contains an element not contained in a given finite subset. (Contributed by Jim Kingdon, 7-Aug-2023.) |
Ref | Expression |
---|---|
inffinp1.dc | ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦) |
inffinp1.inf | ⊢ (𝜑 → ω ≼ 𝐴) |
inffinp1.ss | ⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
inffinp1.b | ⊢ (𝜑 → 𝐵 ∈ Fin) |
Ref | Expression |
---|---|
inffinp1 | ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inffinp1.dc | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦) | |
2 | inffinp1.inf | . . . 4 ⊢ (𝜑 → ω ≼ 𝐴) | |
3 | inffinp1.ss | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) | |
4 | inffinp1.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ Fin) | |
5 | difinfinf 6986 | . . . 4 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ∈ Fin)) → ω ≼ (𝐴 ∖ 𝐵)) | |
6 | 1, 2, 3, 4, 5 | syl22anc 1217 | . . 3 ⊢ (𝜑 → ω ≼ (𝐴 ∖ 𝐵)) |
7 | infm 6798 | . . 3 ⊢ (ω ≼ (𝐴 ∖ 𝐵) → ∃𝑥 𝑥 ∈ (𝐴 ∖ 𝐵)) | |
8 | 6, 7 | syl 14 | . 2 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ (𝐴 ∖ 𝐵)) |
9 | eldif 3080 | . . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) | |
10 | 9 | exbii 1584 | . . 3 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∖ 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) |
11 | df-rex 2422 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) | |
12 | 10, 11 | bitr4i 186 | . 2 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∖ 𝐵) ↔ ∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) |
13 | 8, 12 | sylib 121 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 DECID wdc 819 ∃wex 1468 ∈ wcel 1480 ∀wral 2416 ∃wrex 2417 ∖ cdif 3068 ⊆ wss 3071 class class class wbr 3929 ωcom 4504 ≼ cdom 6633 Fincfn 6634 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-1st 6038 df-2nd 6039 df-1o 6313 df-er 6429 df-en 6635 df-dom 6636 df-fin 6637 df-dju 6923 df-inl 6932 df-inr 6933 df-case 6969 |
This theorem is referenced by: ctinf 11943 |
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