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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 7100 | . . . 4 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ∈ Fin)) → ω ≼ (𝐴 ∖ 𝐵)) | |
6 | 1, 2, 3, 4, 5 | syl22anc 1239 | . . 3 ⊢ (𝜑 → ω ≼ (𝐴 ∖ 𝐵)) |
7 | infm 6904 | . . 3 ⊢ (ω ≼ (𝐴 ∖ 𝐵) → ∃𝑥 𝑥 ∈ (𝐴 ∖ 𝐵)) | |
8 | 6, 7 | syl 14 | . 2 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ (𝐴 ∖ 𝐵)) |
9 | eldif 3139 | . . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) | |
10 | 9 | exbii 1605 | . . 3 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∖ 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) |
11 | df-rex 2461 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) | |
12 | 10, 11 | bitr4i 187 | . 2 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∖ 𝐵) ↔ ∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) |
13 | 8, 12 | sylib 122 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 DECID wdc 834 ∃wex 1492 ∈ wcel 2148 ∀wral 2455 ∃wrex 2456 ∖ cdif 3127 ⊆ wss 3130 class class class wbr 4004 ωcom 4590 ≼ cdom 6739 Fincfn 6740 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4119 ax-sep 4122 ax-nul 4130 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 ax-iinf 4588 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2740 df-sbc 2964 df-csb 3059 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-nul 3424 df-if 3536 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-int 3846 df-iun 3889 df-br 4005 df-opab 4066 df-mpt 4067 df-tr 4103 df-id 4294 df-iord 4367 df-on 4369 df-suc 4372 df-iom 4591 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-f1 5222 df-fo 5223 df-f1o 5224 df-fv 5225 df-1st 6141 df-2nd 6142 df-1o 6417 df-er 6535 df-en 6741 df-dom 6742 df-fin 6743 df-dju 7037 df-inl 7046 df-inr 7047 df-case 7083 |
This theorem is referenced by: ctinf 12431 |
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