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Mirrors > Home > ILE Home > Th. List > infnfi | GIF version |
Description: An infinite set is not finite. (Contributed by Jim Kingdon, 20-Feb-2022.) |
Ref | Expression |
---|---|
infnfi | ⊢ (ω ≼ 𝐴 → ¬ 𝐴 ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfi 6663 | . . . . 5 ⊢ (𝐴 ∈ Fin ↔ ∃𝑛 ∈ ω 𝐴 ≈ 𝑛) | |
2 | 1 | biimpi 119 | . . . 4 ⊢ (𝐴 ∈ Fin → ∃𝑛 ∈ ω 𝐴 ≈ 𝑛) |
3 | 2 | adantl 275 | . . 3 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ Fin) → ∃𝑛 ∈ ω 𝐴 ≈ 𝑛) |
4 | omex 4515 | . . . . . 6 ⊢ ω ∈ V | |
5 | ordom 4528 | . . . . . . 7 ⊢ Ord ω | |
6 | peano2 4517 | . . . . . . . 8 ⊢ (𝑛 ∈ ω → suc 𝑛 ∈ ω) | |
7 | 6 | ad2antrl 482 | . . . . . . 7 ⊢ (((ω ≼ 𝐴 ∧ 𝐴 ∈ Fin) ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → suc 𝑛 ∈ ω) |
8 | ordelss 4309 | . . . . . . 7 ⊢ ((Ord ω ∧ suc 𝑛 ∈ ω) → suc 𝑛 ⊆ ω) | |
9 | 5, 7, 8 | sylancr 411 | . . . . . 6 ⊢ (((ω ≼ 𝐴 ∧ 𝐴 ∈ Fin) ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → suc 𝑛 ⊆ ω) |
10 | ssdomg 6680 | . . . . . 6 ⊢ (ω ∈ V → (suc 𝑛 ⊆ ω → suc 𝑛 ≼ ω)) | |
11 | 4, 9, 10 | mpsyl 65 | . . . . 5 ⊢ (((ω ≼ 𝐴 ∧ 𝐴 ∈ Fin) ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → suc 𝑛 ≼ ω) |
12 | domentr 6693 | . . . . . 6 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ≈ 𝑛) → ω ≼ 𝑛) | |
13 | 12 | ad2ant2rl 503 | . . . . 5 ⊢ (((ω ≼ 𝐴 ∧ 𝐴 ∈ Fin) ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → ω ≼ 𝑛) |
14 | domtr 6687 | . . . . 5 ⊢ ((suc 𝑛 ≼ ω ∧ ω ≼ 𝑛) → suc 𝑛 ≼ 𝑛) | |
15 | 11, 13, 14 | syl2anc 409 | . . . 4 ⊢ (((ω ≼ 𝐴 ∧ 𝐴 ∈ Fin) ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → suc 𝑛 ≼ 𝑛) |
16 | php5dom 6765 | . . . . 5 ⊢ (𝑛 ∈ ω → ¬ suc 𝑛 ≼ 𝑛) | |
17 | 16 | ad2antrl 482 | . . . 4 ⊢ (((ω ≼ 𝐴 ∧ 𝐴 ∈ Fin) ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → ¬ suc 𝑛 ≼ 𝑛) |
18 | 15, 17 | pm2.21dd 610 | . . 3 ⊢ (((ω ≼ 𝐴 ∧ 𝐴 ∈ Fin) ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → ¬ 𝐴 ∈ Fin) |
19 | 3, 18 | rexlimddv 2557 | . 2 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ Fin) → ¬ 𝐴 ∈ Fin) |
20 | 19 | pm2.01da 626 | 1 ⊢ (ω ≼ 𝐴 → ¬ 𝐴 ∈ Fin) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∈ wcel 1481 ∃wrex 2418 Vcvv 2689 ⊆ wss 3076 class class class wbr 3937 Ord word 4292 suc csuc 4295 ωcom 4512 ≈ cen 6640 ≼ cdom 6641 Fincfn 6642 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-tr 4035 df-id 4223 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-er 6437 df-en 6643 df-dom 6644 df-fin 6645 |
This theorem is referenced by: ominf 6798 hashennnuni 10557 |
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