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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 7865 | . . 3 ⊢ (ω ∈ Fin ↔ ∃𝑥 ∈ ω ω ≈ 𝑥) | |
2 | nnord 6965 | . . . . . . . 8 ⊢ (𝑥 ∈ ω → Ord 𝑥) | |
3 | ordom 6966 | . . . . . . . 8 ⊢ Ord ω | |
4 | ordelssne 5667 | . . . . . . . 8 ⊢ ((Ord 𝑥 ∧ Ord ω) → (𝑥 ∈ ω ↔ (𝑥 ⊆ ω ∧ 𝑥 ≠ ω))) | |
5 | 2, 3, 4 | sylancl 693 | . . . . . . 7 ⊢ (𝑥 ∈ ω → (𝑥 ∈ ω ↔ (𝑥 ⊆ ω ∧ 𝑥 ≠ ω))) |
6 | 5 | ibi 255 | . . . . . 6 ⊢ (𝑥 ∈ ω → (𝑥 ⊆ ω ∧ 𝑥 ≠ ω)) |
7 | df-pss 3556 | . . . . . 6 ⊢ (𝑥 ⊊ ω ↔ (𝑥 ⊆ ω ∧ 𝑥 ≠ ω)) | |
8 | 6, 7 | sylibr 223 | . . . . 5 ⊢ (𝑥 ∈ ω → 𝑥 ⊊ ω) |
9 | ensym 7891 | . . . . 5 ⊢ (ω ≈ 𝑥 → 𝑥 ≈ ω) | |
10 | pssinf 8055 | . . . . 5 ⊢ ((𝑥 ⊊ ω ∧ 𝑥 ≈ ω) → ¬ ω ∈ Fin) | |
11 | 8, 9, 10 | syl2an 493 | . . . 4 ⊢ ((𝑥 ∈ ω ∧ ω ≈ 𝑥) → ¬ ω ∈ Fin) |
12 | 11 | rexlimiva 3010 | . . 3 ⊢ (∃𝑥 ∈ ω ω ≈ 𝑥 → ¬ ω ∈ Fin) |
13 | 1, 12 | sylbi 206 | . 2 ⊢ (ω ∈ Fin → ¬ ω ∈ Fin) |
14 | pm2.01 179 | . 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 195 ∧ wa 383 ∈ wcel 1977 ≠ wne 2780 ∃wrex 2897 ⊆ wss 3540 ⊊ wpss 3541 class class class wbr 4583 Ord word 5639 ωcom 6957 ≈ cen 7838 Fincfn 7841 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-om 6958 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 |
This theorem is referenced by: fineqv 8060 nnsdomg 8104 ackbij1lem18 8942 fin23lem21 9044 fin23lem28 9045 fin23lem30 9047 isfin1-2 9090 uzinf 12626 bitsf1 15006 odhash 17812 ufinffr 21543 poimirlem30 32609 diophin 36354 diophren 36395 fiphp3d 36401 |
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