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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 10415 and nsmallnq 10399). (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 3512 | . . . 4 ⊢ (𝐴 ∈ P → 𝐴 ∈ V) | |
2 | prnmax 10417 | . . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦) | |
3 | 2 | ralrimiva 3182 | . . . . 5 ⊢ (𝐴 ∈ P → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦) |
4 | prpssnq 10412 | . . . . . . . . . . 11 ⊢ (𝐴 ∈ P → 𝐴 ⊊ Q) | |
5 | 4 | pssssd 4074 | . . . . . . . . . 10 ⊢ (𝐴 ∈ P → 𝐴 ⊆ Q) |
6 | ltsonq 10391 | . . . . . . . . . 10 ⊢ <Q Or Q | |
7 | soss 5493 | . . . . . . . . . 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 10411 | . . . . . . . . 9 ⊢ (𝐴 ∈ P → 𝐴 ≠ ∅) | |
12 | 11 | adantr 483 | . . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐴 ∈ Fin) → 𝐴 ≠ ∅) |
13 | fimax2g 8764 | . . . . . . . 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 3236 | . . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <Q 𝑦 ↔ ¬ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦) | |
16 | 15 | rexbii 3247 | . . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <Q 𝑦 ↔ ∃𝑥 ∈ 𝐴 ¬ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦) |
17 | rexnal 3238 | . . . . . . . 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 3113 | . . . 4 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐴 ∈ Fin) → V ≠ Fin) | |
23 | 1, 21, 22 | syl2anc 586 | . . 3 ⊢ (𝐴 ∈ P → V ≠ Fin) |
24 | 23 | necomd 3071 | . 2 ⊢ (𝐴 ∈ P → Fin ≠ V) |
25 | fineqv 8733 | . . 3 ⊢ (¬ ω ∈ V ↔ Fin = V) | |
26 | 25 | necon1abii 3064 | . 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 3016 ∀wral 3138 ∃wrex 3139 Vcvv 3494 ⊆ wss 3936 ∅c0 4291 class class class wbr 5066 Or wor 5473 ωcom 7580 Fincfn 8509 Qcnq 10274 <Q cltq 10280 Pcnp 10281 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-omul 8107 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-ni 10294 df-mi 10296 df-lti 10297 df-ltpq 10332 df-enq 10333 df-nq 10334 df-ltnq 10340 df-np 10403 |
This theorem is referenced by: (None) |
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