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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 10417 and nsmallnq 10401). (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 3514 | . . . 4 ⊢ (𝐴 ∈ P → 𝐴 ∈ V) | |
2 | prnmax 10419 | . . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦) | |
3 | 2 | ralrimiva 3184 | . . . . 5 ⊢ (𝐴 ∈ P → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦) |
4 | prpssnq 10414 | . . . . . . . . . . 11 ⊢ (𝐴 ∈ P → 𝐴 ⊊ Q) | |
5 | 4 | pssssd 4076 | . . . . . . . . . 10 ⊢ (𝐴 ∈ P → 𝐴 ⊆ Q) |
6 | ltsonq 10393 | . . . . . . . . . 10 ⊢ <Q Or Q | |
7 | soss 5495 | . . . . . . . . . 10 ⊢ (𝐴 ⊆ Q → ( <Q Or Q → <Q Or 𝐴)) | |
8 | 5, 6, 7 | mpisyl 21 | . . . . . . . . 9 ⊢ (𝐴 ∈ P → <Q Or 𝐴) |
9 | 8 | adantr 483 | . . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐴 ∈ Fin) → <Q Or 𝐴) |
10 | simpr 487 | . . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐴 ∈ Fin) → 𝐴 ∈ Fin) | |
11 | prn0 10413 | . . . . . . . . 9 ⊢ (𝐴 ∈ P → 𝐴 ≠ ∅) | |
12 | 11 | adantr 483 | . . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐴 ∈ Fin) → 𝐴 ≠ ∅) |
13 | fimax2g 8766 | . . . . . . . 8 ⊢ (( <Q Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <Q 𝑦) | |
14 | 9, 10, 12, 13 | syl3anc 1367 | . . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐴 ∈ Fin) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <Q 𝑦) |
15 | ralnex 3238 | . . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <Q 𝑦 ↔ ¬ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦) | |
16 | 15 | rexbii 3249 | . . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <Q 𝑦 ↔ ∃𝑥 ∈ 𝐴 ¬ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦) |
17 | rexnal 3240 | . . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 ¬ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦 ↔ ¬ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦) | |
18 | 16, 17 | bitri 277 | . . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <Q 𝑦 ↔ ¬ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦) |
19 | 14, 18 | sylib 220 | . . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐴 ∈ Fin) → ¬ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦) |
20 | 19 | ex 415 | . . . . 5 ⊢ (𝐴 ∈ P → (𝐴 ∈ Fin → ¬ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)) |
21 | 3, 20 | mt2d 138 | . . . 4 ⊢ (𝐴 ∈ P → ¬ 𝐴 ∈ Fin) |
22 | nelne1 3115 | . . . 4 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐴 ∈ Fin) → V ≠ Fin) | |
23 | 1, 21, 22 | syl2anc 586 | . . 3 ⊢ (𝐴 ∈ P → V ≠ Fin) |
24 | 23 | necomd 3073 | . 2 ⊢ (𝐴 ∈ P → Fin ≠ V) |
25 | fineqv 8735 | . . 3 ⊢ (¬ ω ∈ V ↔ Fin = V) | |
26 | 25 | necon1abii 3066 | . 2 ⊢ (Fin ≠ V ↔ ω ∈ V) |
27 | 24, 26 | sylib 220 | 1 ⊢ (𝐴 ∈ P → ω ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 ∃wrex 3141 Vcvv 3496 ⊆ wss 3938 ∅c0 4293 class class class wbr 5068 Or wor 5475 ωcom 7582 Fincfn 8511 Qcnq 10276 <Q cltq 10282 Pcnp 10283 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-omul 8109 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-ni 10296 df-mi 10298 df-lti 10299 df-ltpq 10334 df-enq 10335 df-nq 10336 df-ltnq 10342 df-np 10405 |
This theorem is referenced by: (None) |
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