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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 6408 | . . . . 5 ⊢ (𝐴 ∈ Fin ↔ ∃𝑛 ∈ ω 𝐴 ≈ 𝑛) | |
2 | 1 | biimpi 118 | . . . 4 ⊢ (𝐴 ∈ Fin → ∃𝑛 ∈ ω 𝐴 ≈ 𝑛) |
3 | 2 | adantl 271 | . . 3 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ Fin) → ∃𝑛 ∈ ω 𝐴 ≈ 𝑛) |
4 | omex 4371 | . . . . . 6 ⊢ ω ∈ V | |
5 | ordom 4384 | . . . . . . 7 ⊢ Ord ω | |
6 | peano2 4373 | . . . . . . . 8 ⊢ (𝑛 ∈ ω → suc 𝑛 ∈ ω) | |
7 | 6 | ad2antrl 474 | . . . . . . 7 ⊢ (((ω ≼ 𝐴 ∧ 𝐴 ∈ Fin) ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → suc 𝑛 ∈ ω) |
8 | ordelss 4170 | . . . . . . 7 ⊢ ((Ord ω ∧ suc 𝑛 ∈ ω) → suc 𝑛 ⊆ ω) | |
9 | 5, 7, 8 | sylancr 405 | . . . . . 6 ⊢ (((ω ≼ 𝐴 ∧ 𝐴 ∈ Fin) ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → suc 𝑛 ⊆ ω) |
10 | ssdomg 6425 | . . . . . 6 ⊢ (ω ∈ V → (suc 𝑛 ⊆ ω → suc 𝑛 ≼ ω)) | |
11 | 4, 9, 10 | mpsyl 64 | . . . . 5 ⊢ (((ω ≼ 𝐴 ∧ 𝐴 ∈ Fin) ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → suc 𝑛 ≼ ω) |
12 | domentr 6438 | . . . . . 6 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ≈ 𝑛) → ω ≼ 𝑛) | |
13 | 12 | ad2ant2rl 495 | . . . . 5 ⊢ (((ω ≼ 𝐴 ∧ 𝐴 ∈ Fin) ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → ω ≼ 𝑛) |
14 | domtr 6432 | . . . . 5 ⊢ ((suc 𝑛 ≼ ω ∧ ω ≼ 𝑛) → suc 𝑛 ≼ 𝑛) | |
15 | 11, 13, 14 | syl2anc 403 | . . . 4 ⊢ (((ω ≼ 𝐴 ∧ 𝐴 ∈ Fin) ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → suc 𝑛 ≼ 𝑛) |
16 | php5dom 6509 | . . . . 5 ⊢ (𝑛 ∈ ω → ¬ suc 𝑛 ≼ 𝑛) | |
17 | 16 | ad2antrl 474 | . . . 4 ⊢ (((ω ≼ 𝐴 ∧ 𝐴 ∈ Fin) ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → ¬ suc 𝑛 ≼ 𝑛) |
18 | 15, 17 | pm2.21dd 583 | . . 3 ⊢ (((ω ≼ 𝐴 ∧ 𝐴 ∈ Fin) ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → ¬ 𝐴 ∈ Fin) |
19 | 3, 18 | rexlimddv 2487 | . 2 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ Fin) → ¬ 𝐴 ∈ Fin) |
20 | 19 | pm2.01da 598 | 1 ⊢ (ω ≼ 𝐴 → ¬ 𝐴 ∈ Fin) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 ∈ wcel 1434 ∃wrex 2354 Vcvv 2612 ⊆ wss 2984 class class class wbr 3811 Ord word 4153 suc csuc 4156 ωcom 4368 ≈ cen 6385 ≼ cdom 6386 Fincfn 6387 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-nul 3930 ax-pow 3974 ax-pr 4000 ax-un 4224 ax-setind 4316 ax-iinf 4366 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-ral 2358 df-rex 2359 df-rab 2362 df-v 2614 df-sbc 2827 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-nul 3270 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-br 3812 df-opab 3866 df-tr 3902 df-id 4084 df-iord 4157 df-on 4159 df-suc 4162 df-iom 4369 df-xp 4407 df-rel 4408 df-cnv 4409 df-co 4410 df-dm 4411 df-rn 4412 df-res 4413 df-ima 4414 df-iota 4934 df-fun 4971 df-fn 4972 df-f 4973 df-f1 4974 df-fo 4975 df-f1o 4976 df-fv 4977 df-er 6222 df-en 6388 df-dom 6389 df-fin 6390 |
This theorem is referenced by: ominf 6542 hashennnuni 10022 |
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