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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 6802 | . . . . 5 ⊢ (𝐴 ∈ Fin ↔ ∃𝑛 ∈ ω 𝐴 ≈ 𝑛) | |
2 | 1 | biimpi 120 | . . . 4 ⊢ (𝐴 ∈ Fin → ∃𝑛 ∈ ω 𝐴 ≈ 𝑛) |
3 | 2 | adantl 277 | . . 3 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ Fin) → ∃𝑛 ∈ ω 𝐴 ≈ 𝑛) |
4 | omex 4617 | . . . . . 6 ⊢ ω ∈ V | |
5 | ordom 4631 | . . . . . . 7 ⊢ Ord ω | |
6 | peano2 4619 | . . . . . . . 8 ⊢ (𝑛 ∈ ω → suc 𝑛 ∈ ω) | |
7 | 6 | ad2antrl 490 | . . . . . . 7 ⊢ (((ω ≼ 𝐴 ∧ 𝐴 ∈ Fin) ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → suc 𝑛 ∈ ω) |
8 | ordelss 4404 | . . . . . . 7 ⊢ ((Ord ω ∧ suc 𝑛 ∈ ω) → suc 𝑛 ⊆ ω) | |
9 | 5, 7, 8 | sylancr 414 | . . . . . 6 ⊢ (((ω ≼ 𝐴 ∧ 𝐴 ∈ Fin) ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → suc 𝑛 ⊆ ω) |
10 | ssdomg 6819 | . . . . . 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 6832 | . . . . . 6 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ≈ 𝑛) → ω ≼ 𝑛) | |
13 | 12 | ad2ant2rl 511 | . . . . 5 ⊢ (((ω ≼ 𝐴 ∧ 𝐴 ∈ Fin) ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → ω ≼ 𝑛) |
14 | domtr 6826 | . . . . 5 ⊢ ((suc 𝑛 ≼ ω ∧ ω ≼ 𝑛) → suc 𝑛 ≼ 𝑛) | |
15 | 11, 13, 14 | syl2anc 411 | . . . 4 ⊢ (((ω ≼ 𝐴 ∧ 𝐴 ∈ Fin) ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → suc 𝑛 ≼ 𝑛) |
16 | php5dom 6906 | . . . . 5 ⊢ (𝑛 ∈ ω → ¬ suc 𝑛 ≼ 𝑛) | |
17 | 16 | ad2antrl 490 | . . . 4 ⊢ (((ω ≼ 𝐴 ∧ 𝐴 ∈ Fin) ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → ¬ suc 𝑛 ≼ 𝑛) |
18 | 15, 17 | pm2.21dd 621 | . . 3 ⊢ (((ω ≼ 𝐴 ∧ 𝐴 ∈ Fin) ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → ¬ 𝐴 ∈ Fin) |
19 | 3, 18 | rexlimddv 2612 | . 2 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ Fin) → ¬ 𝐴 ∈ Fin) |
20 | 19 | pm2.01da 637 | 1 ⊢ (ω ≼ 𝐴 → ¬ 𝐴 ∈ Fin) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ∈ wcel 2160 ∃wrex 2469 Vcvv 2756 ⊆ wss 3149 class class class wbr 4025 Ord word 4387 suc csuc 4390 ωcom 4614 ≈ cen 6779 ≼ cdom 6780 Fincfn 6781 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4143 ax-nul 4151 ax-pow 4199 ax-pr 4234 ax-un 4458 ax-setind 4561 ax-iinf 4612 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2758 df-sbc 2982 df-dif 3151 df-un 3153 df-in 3155 df-ss 3162 df-nul 3443 df-pw 3599 df-sn 3620 df-pr 3621 df-op 3623 df-uni 3832 df-int 3867 df-br 4026 df-opab 4087 df-tr 4124 df-id 4318 df-iord 4391 df-on 4393 df-suc 4396 df-iom 4615 df-xp 4657 df-rel 4658 df-cnv 4659 df-co 4660 df-dm 4661 df-rn 4662 df-res 4663 df-ima 4664 df-iota 5203 df-fun 5244 df-fn 5245 df-f 5246 df-f1 5247 df-fo 5248 df-f1o 5249 df-fv 5250 df-er 6574 df-en 6782 df-dom 6783 df-fin 6784 |
This theorem is referenced by: ominf 6939 hashennnuni 10824 |
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