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Mirrors > Home > MPE Home > Th. List > npomex | Structured version Visualization version GIF version |
Description: A simplifying observation, and an indication of why any attempt to develop a theory of the real numbers without the Axiom of Infinity is doomed to failure: since every member of P is an infinite set, the negation of Infinity implies that P, and hence ℝ, is empty. (Note that this proof, which used the fact that Dedekind cuts have no maximum, could just as well have used that they have no minimum, since they are downward-closed by prcdnq 10936 and nsmallnq 10920). (Contributed by Mario Carneiro, 11-May-2013.) (Revised by Mario Carneiro, 16-Nov-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
npomex | ⊢ (𝐴 ∈ P → ω ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3466 | . . . 4 ⊢ (𝐴 ∈ P → 𝐴 ∈ V) | |
2 | prnmax 10938 | . . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦) | |
3 | 2 | ralrimiva 3144 | . . . . 5 ⊢ (𝐴 ∈ P → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦) |
4 | prpssnq 10933 | . . . . . . . . . . 11 ⊢ (𝐴 ∈ P → 𝐴 ⊊ Q) | |
5 | 4 | pssssd 4062 | . . . . . . . . . 10 ⊢ (𝐴 ∈ P → 𝐴 ⊆ Q) |
6 | ltsonq 10912 | . . . . . . . . . 10 ⊢ <Q Or Q | |
7 | soss 5570 | . . . . . . . . . 10 ⊢ (𝐴 ⊆ Q → ( <Q Or Q → <Q Or 𝐴)) | |
8 | 5, 6, 7 | mpisyl 21 | . . . . . . . . 9 ⊢ (𝐴 ∈ P → <Q Or 𝐴) |
9 | 8 | adantr 482 | . . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐴 ∈ Fin) → <Q Or 𝐴) |
10 | simpr 486 | . . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐴 ∈ Fin) → 𝐴 ∈ Fin) | |
11 | prn0 10932 | . . . . . . . . 9 ⊢ (𝐴 ∈ P → 𝐴 ≠ ∅) | |
12 | 11 | adantr 482 | . . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐴 ∈ Fin) → 𝐴 ≠ ∅) |
13 | fimax2g 9240 | . . . . . . . 8 ⊢ (( <Q Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <Q 𝑦) | |
14 | 9, 10, 12, 13 | syl3anc 1372 | . . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐴 ∈ Fin) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <Q 𝑦) |
15 | ralnex 3076 | . . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <Q 𝑦 ↔ ¬ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦) | |
16 | 15 | rexbii 3098 | . . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <Q 𝑦 ↔ ∃𝑥 ∈ 𝐴 ¬ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦) |
17 | rexnal 3104 | . . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 ¬ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦 ↔ ¬ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦) | |
18 | 16, 17 | bitri 275 | . . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <Q 𝑦 ↔ ¬ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦) |
19 | 14, 18 | sylib 217 | . . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐴 ∈ Fin) → ¬ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦) |
20 | 19 | ex 414 | . . . . 5 ⊢ (𝐴 ∈ P → (𝐴 ∈ Fin → ¬ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)) |
21 | 3, 20 | mt2d 136 | . . . 4 ⊢ (𝐴 ∈ P → ¬ 𝐴 ∈ Fin) |
22 | nelne1 3042 | . . . 4 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐴 ∈ Fin) → V ≠ Fin) | |
23 | 1, 21, 22 | syl2anc 585 | . . 3 ⊢ (𝐴 ∈ P → V ≠ Fin) |
24 | 23 | necomd 3000 | . 2 ⊢ (𝐴 ∈ P → Fin ≠ V) |
25 | fineqv 9214 | . . 3 ⊢ (¬ ω ∈ V ↔ Fin = V) | |
26 | 25 | necon1abii 2993 | . 2 ⊢ (Fin ≠ V ↔ ω ∈ V) |
27 | 24, 26 | sylib 217 | 1 ⊢ (𝐴 ∈ P → ω ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 ∈ wcel 2107 ≠ wne 2944 ∀wral 3065 ∃wrex 3074 Vcvv 3448 ⊆ wss 3915 ∅c0 4287 class class class wbr 5110 Or wor 5549 ωcom 7807 Fincfn 8890 Qcnq 10795 <Q cltq 10801 Pcnp 10802 |
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-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 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-rmo 3356 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-iun 4961 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-pred 6258 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-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-oadd 8421 df-omul 8422 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-ni 10815 df-mi 10817 df-lti 10818 df-ltpq 10853 df-enq 10854 df-nq 10855 df-ltnq 10861 df-np 10924 |
This theorem is referenced by: (None) |
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