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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 10404 and nsmallnq 10388). (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 3459 | . . . 4 ⊢ (𝐴 ∈ P → 𝐴 ∈ V) | |
2 | prnmax 10406 | . . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦) | |
3 | 2 | ralrimiva 3149 | . . . . 5 ⊢ (𝐴 ∈ P → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦) |
4 | prpssnq 10401 | . . . . . . . . . . 11 ⊢ (𝐴 ∈ P → 𝐴 ⊊ Q) | |
5 | 4 | pssssd 4025 | . . . . . . . . . 10 ⊢ (𝐴 ∈ P → 𝐴 ⊆ Q) |
6 | ltsonq 10380 | . . . . . . . . . 10 ⊢ <Q Or Q | |
7 | soss 5457 | . . . . . . . . . 10 ⊢ (𝐴 ⊆ Q → ( <Q Or Q → <Q Or 𝐴)) | |
8 | 5, 6, 7 | mpisyl 21 | . . . . . . . . 9 ⊢ (𝐴 ∈ P → <Q Or 𝐴) |
9 | 8 | adantr 484 | . . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐴 ∈ Fin) → <Q Or 𝐴) |
10 | simpr 488 | . . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐴 ∈ Fin) → 𝐴 ∈ Fin) | |
11 | prn0 10400 | . . . . . . . . 9 ⊢ (𝐴 ∈ P → 𝐴 ≠ ∅) | |
12 | 11 | adantr 484 | . . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐴 ∈ Fin) → 𝐴 ≠ ∅) |
13 | fimax2g 8748 | . . . . . . . 8 ⊢ (( <Q Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <Q 𝑦) | |
14 | 9, 10, 12, 13 | syl3anc 1368 | . . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐴 ∈ Fin) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <Q 𝑦) |
15 | ralnex 3199 | . . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <Q 𝑦 ↔ ¬ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦) | |
16 | 15 | rexbii 3210 | . . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <Q 𝑦 ↔ ∃𝑥 ∈ 𝐴 ¬ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦) |
17 | rexnal 3201 | . . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 ¬ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦 ↔ ¬ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦) | |
18 | 16, 17 | bitri 278 | . . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <Q 𝑦 ↔ ¬ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦) |
19 | 14, 18 | sylib 221 | . . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐴 ∈ Fin) → ¬ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦) |
20 | 19 | ex 416 | . . . . 5 ⊢ (𝐴 ∈ P → (𝐴 ∈ Fin → ¬ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)) |
21 | 3, 20 | mt2d 138 | . . . 4 ⊢ (𝐴 ∈ P → ¬ 𝐴 ∈ Fin) |
22 | nelne1 3083 | . . . 4 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐴 ∈ Fin) → V ≠ Fin) | |
23 | 1, 21, 22 | syl2anc 587 | . . 3 ⊢ (𝐴 ∈ P → V ≠ Fin) |
24 | 23 | necomd 3042 | . 2 ⊢ (𝐴 ∈ P → Fin ≠ V) |
25 | fineqv 8717 | . . 3 ⊢ (¬ ω ∈ V ↔ Fin = V) | |
26 | 25 | necon1abii 3035 | . 2 ⊢ (Fin ≠ V ↔ ω ∈ V) |
27 | 24, 26 | sylib 221 | 1 ⊢ (𝐴 ∈ P → ω ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ∃wrex 3107 Vcvv 3441 ⊆ wss 3881 ∅c0 4243 class class class wbr 5030 Or wor 5437 ωcom 7560 Fincfn 8492 Qcnq 10263 <Q cltq 10269 Pcnp 10270 |
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-rep 5154 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-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 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-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 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-pred 6116 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-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-omul 8090 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-ni 10283 df-mi 10285 df-lti 10286 df-ltpq 10321 df-enq 10322 df-nq 10323 df-ltnq 10329 df-np 10392 |
This theorem is referenced by: (None) |
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