![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ominf | Structured version Visualization version GIF version |
Description: The set of natural numbers is infinite. Corollary 6D(b) of [Enderton] p. 136. (Contributed by NM, 2-Jun-1998.) Avoid ax-pow 5325. (Revised by BTernaryTau, 2-Jan-2025.) |
Ref | Expression |
---|---|
ominf | ⊢ ¬ ω ∈ Fin |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfi 8923 | . . 3 ⊢ (ω ∈ Fin ↔ ∃𝑥 ∈ ω ω ≈ 𝑥) | |
2 | nnord 7815 | . . . . . . . 8 ⊢ (𝑥 ∈ ω → Ord 𝑥) | |
3 | ordom 7817 | . . . . . . . 8 ⊢ Ord ω | |
4 | ordelssne 6349 | . . . . . . . 8 ⊢ ((Ord 𝑥 ∧ Ord ω) → (𝑥 ∈ ω ↔ (𝑥 ⊆ ω ∧ 𝑥 ≠ ω))) | |
5 | 2, 3, 4 | sylancl 587 | . . . . . . 7 ⊢ (𝑥 ∈ ω → (𝑥 ∈ ω ↔ (𝑥 ⊆ ω ∧ 𝑥 ≠ ω))) |
6 | 5 | ibi 267 | . . . . . 6 ⊢ (𝑥 ∈ ω → (𝑥 ⊆ ω ∧ 𝑥 ≠ ω)) |
7 | df-pss 3934 | . . . . . 6 ⊢ (𝑥 ⊊ ω ↔ (𝑥 ⊆ ω ∧ 𝑥 ≠ ω)) | |
8 | 6, 7 | sylibr 233 | . . . . 5 ⊢ (𝑥 ∈ ω → 𝑥 ⊊ ω) |
9 | nnfi 9118 | . . . . . . 7 ⊢ (𝑥 ∈ ω → 𝑥 ∈ Fin) | |
10 | ensymfib 9138 | . . . . . . 7 ⊢ (𝑥 ∈ Fin → (𝑥 ≈ ω ↔ ω ≈ 𝑥)) | |
11 | 9, 10 | syl 17 | . . . . . 6 ⊢ (𝑥 ∈ ω → (𝑥 ≈ ω ↔ ω ≈ 𝑥)) |
12 | 11 | biimpar 479 | . . . . 5 ⊢ ((𝑥 ∈ ω ∧ ω ≈ 𝑥) → 𝑥 ≈ ω) |
13 | pssinf 9207 | . . . . 5 ⊢ ((𝑥 ⊊ ω ∧ 𝑥 ≈ ω) → ¬ ω ∈ Fin) | |
14 | 8, 12, 13 | syl2an2r 684 | . . . 4 ⊢ ((𝑥 ∈ ω ∧ ω ≈ 𝑥) → ¬ ω ∈ Fin) |
15 | 14 | rexlimiva 3145 | . . 3 ⊢ (∃𝑥 ∈ ω ω ≈ 𝑥 → ¬ ω ∈ Fin) |
16 | 1, 15 | sylbi 216 | . 2 ⊢ (ω ∈ Fin → ¬ ω ∈ Fin) |
17 | pm2.01 188 | . 2 ⊢ ((ω ∈ Fin → ¬ ω ∈ Fin) → ¬ ω ∈ Fin) | |
18 | 16, 17 | ax-mp 5 | 1 ⊢ ¬ ω ∈ Fin |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∈ wcel 2107 ≠ wne 2944 ∃wrex 3074 ⊆ wss 3915 ⊊ wpss 3916 class class class wbr 5110 Ord word 6321 ωcom 7807 ≈ cen 8887 Fincfn 8890 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-om 7808 df-1o 8417 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 |
This theorem is referenced by: fineqv 9214 nnsdomg 9253 nnsdomgOLD 9254 ackbij1lem18 10180 fin23lem21 10282 fin23lem28 10283 fin23lem30 10285 isfin1-2 10328 uzinf 13877 bitsf1 16333 odhash 19363 ufinffr 23296 poimirlem30 36137 diophin 41124 diophren 41165 fiphp3d 41171 |
Copyright terms: Public domain | W3C validator |