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Mirrors > Home > MPE Home > Th. List > qnnen | Structured version Visualization version GIF version |
Description: The rational numbers are countable. This proof does not use the Axiom of Choice, even though it uses an onto function, because the base set (ℤ × ℕ) is numerable. Exercise 2 of [Enderton] p. 133. For purposes of the Metamath 100 list, we are considering Mario Carneiro's revision as the date this proof was completed. This is Metamath 100 proof #3. (Contributed by NM, 31-Jul-2004.) (Revised by Mario Carneiro, 3-Mar-2013.) |
Ref | Expression |
---|---|
qnnen | ⊢ ℚ ≈ ℕ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omelon 9109 | . . . . . . 7 ⊢ ω ∈ On | |
2 | nnenom 13349 | . . . . . . . 8 ⊢ ℕ ≈ ω | |
3 | 2 | ensymi 8559 | . . . . . . 7 ⊢ ω ≈ ℕ |
4 | isnumi 9375 | . . . . . . 7 ⊢ ((ω ∈ On ∧ ω ≈ ℕ) → ℕ ∈ dom card) | |
5 | 1, 3, 4 | mp2an 690 | . . . . . 6 ⊢ ℕ ∈ dom card |
6 | znnen 15565 | . . . . . . 7 ⊢ ℤ ≈ ℕ | |
7 | ennum 9376 | . . . . . . 7 ⊢ (ℤ ≈ ℕ → (ℤ ∈ dom card ↔ ℕ ∈ dom card)) | |
8 | 6, 7 | ax-mp 5 | . . . . . 6 ⊢ (ℤ ∈ dom card ↔ ℕ ∈ dom card) |
9 | 5, 8 | mpbir 233 | . . . . 5 ⊢ ℤ ∈ dom card |
10 | xpnum 9380 | . . . . 5 ⊢ ((ℤ ∈ dom card ∧ ℕ ∈ dom card) → (ℤ × ℕ) ∈ dom card) | |
11 | 9, 5, 10 | mp2an 690 | . . . 4 ⊢ (ℤ × ℕ) ∈ dom card |
12 | eqid 2821 | . . . . . 6 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) | |
13 | ovex 7189 | . . . . . 6 ⊢ (𝑥 / 𝑦) ∈ V | |
14 | 12, 13 | fnmpoi 7768 | . . . . 5 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) Fn (ℤ × ℕ) |
15 | 12 | rnmpo 7284 | . . . . . 6 ⊢ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦)} |
16 | elq 12351 | . . . . . . 7 ⊢ (𝑧 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦)) | |
17 | 16 | abbi2i 2953 | . . . . . 6 ⊢ ℚ = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦)} |
18 | 15, 17 | eqtr4i 2847 | . . . . 5 ⊢ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = ℚ |
19 | df-fo 6361 | . . . . 5 ⊢ ((𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)):(ℤ × ℕ)–onto→ℚ ↔ ((𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) Fn (ℤ × ℕ) ∧ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = ℚ)) | |
20 | 14, 18, 19 | mpbir2an 709 | . . . 4 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)):(ℤ × ℕ)–onto→ℚ |
21 | fodomnum 9483 | . . . 4 ⊢ ((ℤ × ℕ) ∈ dom card → ((𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)):(ℤ × ℕ)–onto→ℚ → ℚ ≼ (ℤ × ℕ))) | |
22 | 11, 20, 21 | mp2 9 | . . 3 ⊢ ℚ ≼ (ℤ × ℕ) |
23 | nnex 11644 | . . . . . 6 ⊢ ℕ ∈ V | |
24 | 23 | enref 8542 | . . . . 5 ⊢ ℕ ≈ ℕ |
25 | xpen 8680 | . . . . 5 ⊢ ((ℤ ≈ ℕ ∧ ℕ ≈ ℕ) → (ℤ × ℕ) ≈ (ℕ × ℕ)) | |
26 | 6, 24, 25 | mp2an 690 | . . . 4 ⊢ (ℤ × ℕ) ≈ (ℕ × ℕ) |
27 | xpnnen 15564 | . . . 4 ⊢ (ℕ × ℕ) ≈ ℕ | |
28 | 26, 27 | entri 8563 | . . 3 ⊢ (ℤ × ℕ) ≈ ℕ |
29 | domentr 8568 | . . 3 ⊢ ((ℚ ≼ (ℤ × ℕ) ∧ (ℤ × ℕ) ≈ ℕ) → ℚ ≼ ℕ) | |
30 | 22, 28, 29 | mp2an 690 | . 2 ⊢ ℚ ≼ ℕ |
31 | qex 12361 | . . 3 ⊢ ℚ ∈ V | |
32 | nnssq 12358 | . . 3 ⊢ ℕ ⊆ ℚ | |
33 | ssdomg 8555 | . . 3 ⊢ (ℚ ∈ V → (ℕ ⊆ ℚ → ℕ ≼ ℚ)) | |
34 | 31, 32, 33 | mp2 9 | . 2 ⊢ ℕ ≼ ℚ |
35 | sbth 8637 | . 2 ⊢ ((ℚ ≼ ℕ ∧ ℕ ≼ ℚ) → ℚ ≈ ℕ) | |
36 | 30, 34, 35 | mp2an 690 | 1 ⊢ ℚ ≈ ℕ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∈ wcel 2114 {cab 2799 ∃wrex 3139 Vcvv 3494 ⊆ wss 3936 class class class wbr 5066 × cxp 5553 dom cdm 5555 ran crn 5556 Oncon0 6191 Fn wfn 6350 –onto→wfo 6353 (class class class)co 7156 ∈ cmpo 7158 ωcom 7580 ≈ cen 8506 ≼ cdom 8507 cardccrd 9364 / cdiv 11297 ℕcn 11638 ℤcz 11982 ℚcq 12349 |
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 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
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-nel 3124 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-int 4877 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-se 5515 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-isom 6364 df-riota 7114 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-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-oi 8974 df-card 9368 df-acn 9371 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-q 12350 |
This theorem is referenced by: rpnnen 15580 resdomq 15597 re2ndc 23409 ovolq 24092 opnmblALT 24204 vitali 24214 mbfimaopnlem 24256 mbfaddlem 24261 mblfinlem1 34944 irrapx1 39445 qenom 41649 |
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