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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > nnfoctb | Structured version Visualization version GIF version |
Description: There exists a mapping from ℕ onto any (nonempty) countable set. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
nnfoctb | ⊢ ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → ∃𝑓 𝑓:ℕ–onto→𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 483 | . . 3 ⊢ ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → 𝐴 ≠ ∅) | |
2 | reldom 8976 | . . . . . . 7 ⊢ Rel ≼ | |
3 | 2 | a1i 11 | . . . . . 6 ⊢ (𝐴 ≼ ω → Rel ≼ ) |
4 | brrelex1 5735 | . . . . . 6 ⊢ ((Rel ≼ ∧ 𝐴 ≼ ω) → 𝐴 ∈ V) | |
5 | 3, 4 | mpancom 686 | . . . . 5 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V) |
6 | 0sdomg 9135 | . . . . 5 ⊢ (𝐴 ∈ V → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ (𝐴 ≼ ω → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) |
8 | 7 | adantr 479 | . . 3 ⊢ ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) |
9 | 1, 8 | mpbird 256 | . 2 ⊢ ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → ∅ ≺ 𝐴) |
10 | nnenom 13985 | . . . . . 6 ⊢ ℕ ≈ ω | |
11 | 10 | ensymi 9031 | . . . . 5 ⊢ ω ≈ ℕ |
12 | 11 | a1i 11 | . . . 4 ⊢ (𝐴 ≼ ω → ω ≈ ℕ) |
13 | domentr 9040 | . . . 4 ⊢ ((𝐴 ≼ ω ∧ ω ≈ ℕ) → 𝐴 ≼ ℕ) | |
14 | 12, 13 | mpdan 685 | . . 3 ⊢ (𝐴 ≼ ω → 𝐴 ≼ ℕ) |
15 | 14 | adantr 479 | . 2 ⊢ ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → 𝐴 ≼ ℕ) |
16 | fodomr 9159 | . 2 ⊢ ((∅ ≺ 𝐴 ∧ 𝐴 ≼ ℕ) → ∃𝑓 𝑓:ℕ–onto→𝐴) | |
17 | 9, 15, 16 | syl2anc 582 | 1 ⊢ ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → ∃𝑓 𝑓:ℕ–onto→𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∃wex 1773 ∈ wcel 2098 ≠ wne 2937 Vcvv 3473 ∅c0 4326 class class class wbr 5152 Rel wrel 5687 –onto→wfo 6551 ωcom 7876 ≈ cen 8967 ≼ cdom 8968 ≺ csdm 8969 ℕcn 12250 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-n0 12511 df-z 12597 df-uz 12861 |
This theorem is referenced by: ssnnf1octb 44597 issalnnd 45762 nnfoctbdj 45873 |
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