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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 6739 | . . . . 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 4577 | . . . . . 6 ⊢ ω ∈ V | |
5 | ordom 4591 | . . . . . . 7 ⊢ Ord ω | |
6 | peano2 4579 | . . . . . . . 8 ⊢ (𝑛 ∈ ω → suc 𝑛 ∈ ω) | |
7 | 6 | ad2antrl 487 | . . . . . . 7 ⊢ (((ω ≼ 𝐴 ∧ 𝐴 ∈ Fin) ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → suc 𝑛 ∈ ω) |
8 | ordelss 4364 | . . . . . . 7 ⊢ ((Ord ω ∧ suc 𝑛 ∈ ω) → suc 𝑛 ⊆ ω) | |
9 | 5, 7, 8 | sylancr 412 | . . . . . 6 ⊢ (((ω ≼ 𝐴 ∧ 𝐴 ∈ Fin) ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → suc 𝑛 ⊆ ω) |
10 | ssdomg 6756 | . . . . . 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 6769 | . . . . . 6 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ≈ 𝑛) → ω ≼ 𝑛) | |
13 | 12 | ad2ant2rl 508 | . . . . 5 ⊢ (((ω ≼ 𝐴 ∧ 𝐴 ∈ Fin) ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → ω ≼ 𝑛) |
14 | domtr 6763 | . . . . 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 6841 | . . . . 5 ⊢ (𝑛 ∈ ω → ¬ suc 𝑛 ≼ 𝑛) | |
17 | 16 | ad2antrl 487 | . . . 4 ⊢ (((ω ≼ 𝐴 ∧ 𝐴 ∈ Fin) ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → ¬ suc 𝑛 ≼ 𝑛) |
18 | 15, 17 | pm2.21dd 615 | . . 3 ⊢ (((ω ≼ 𝐴 ∧ 𝐴 ∈ Fin) ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → ¬ 𝐴 ∈ Fin) |
19 | 3, 18 | rexlimddv 2592 | . 2 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ Fin) → ¬ 𝐴 ∈ Fin) |
20 | 19 | pm2.01da 631 | 1 ⊢ (ω ≼ 𝐴 → ¬ 𝐴 ∈ Fin) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∈ wcel 2141 ∃wrex 2449 Vcvv 2730 ⊆ wss 3121 class class class wbr 3989 Ord word 4347 suc csuc 4350 ωcom 4574 ≈ cen 6716 ≼ cdom 6717 Fincfn 6718 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-er 6513 df-en 6719 df-dom 6720 df-fin 6721 |
This theorem is referenced by: ominf 6874 hashennnuni 10713 |
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