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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.) |
Ref | Expression |
---|---|
ominf | ⊢ ¬ ω ∈ Fin |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfi 8516 | . . 3 ⊢ (ω ∈ Fin ↔ ∃𝑥 ∈ ω ω ≈ 𝑥) | |
2 | nnord 7568 | . . . . . . . 8 ⊢ (𝑥 ∈ ω → Ord 𝑥) | |
3 | ordom 7569 | . . . . . . . 8 ⊢ Ord ω | |
4 | ordelssne 6186 | . . . . . . . 8 ⊢ ((Ord 𝑥 ∧ Ord ω) → (𝑥 ∈ ω ↔ (𝑥 ⊆ ω ∧ 𝑥 ≠ ω))) | |
5 | 2, 3, 4 | sylancl 589 | . . . . . . 7 ⊢ (𝑥 ∈ ω → (𝑥 ∈ ω ↔ (𝑥 ⊆ ω ∧ 𝑥 ≠ ω))) |
6 | 5 | ibi 270 | . . . . . 6 ⊢ (𝑥 ∈ ω → (𝑥 ⊆ ω ∧ 𝑥 ≠ ω)) |
7 | df-pss 3900 | . . . . . 6 ⊢ (𝑥 ⊊ ω ↔ (𝑥 ⊆ ω ∧ 𝑥 ≠ ω)) | |
8 | 6, 7 | sylibr 237 | . . . . 5 ⊢ (𝑥 ∈ ω → 𝑥 ⊊ ω) |
9 | ensym 8541 | . . . . 5 ⊢ (ω ≈ 𝑥 → 𝑥 ≈ ω) | |
10 | pssinf 8712 | . . . . 5 ⊢ ((𝑥 ⊊ ω ∧ 𝑥 ≈ ω) → ¬ ω ∈ Fin) | |
11 | 8, 9, 10 | syl2an 598 | . . . 4 ⊢ ((𝑥 ∈ ω ∧ ω ≈ 𝑥) → ¬ ω ∈ Fin) |
12 | 11 | rexlimiva 3240 | . . 3 ⊢ (∃𝑥 ∈ ω ω ≈ 𝑥 → ¬ ω ∈ Fin) |
13 | 1, 12 | sylbi 220 | . 2 ⊢ (ω ∈ Fin → ¬ ω ∈ Fin) |
14 | pm2.01 192 | . 2 ⊢ ((ω ∈ Fin → ¬ ω ∈ Fin) → ¬ ω ∈ Fin) | |
15 | 13, 14 | ax-mp 5 | 1 ⊢ ¬ ω ∈ Fin |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2111 ≠ wne 2987 ∃wrex 3107 ⊆ wss 3881 ⊊ wpss 3882 class class class wbr 5030 Ord word 6158 ωcom 7560 ≈ cen 8489 Fincfn 8492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-om 7561 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 |
This theorem is referenced by: fineqv 8717 nnsdomg 8761 ackbij1lem18 9648 fin23lem21 9750 fin23lem28 9751 fin23lem30 9753 isfin1-2 9796 uzinf 13328 bitsf1 15785 odhash 18691 ufinffr 22534 poimirlem30 35087 diophin 39713 diophren 39754 fiphp3d 39760 |
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