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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 488 | . . 3 ⊢ ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → 𝐴 ≠ ∅) | |
2 | reldom 8498 | . . . . . . 7 ⊢ Rel ≼ | |
3 | 2 | a1i 11 | . . . . . 6 ⊢ (𝐴 ≼ ω → Rel ≼ ) |
4 | brrelex1 5569 | . . . . . 6 ⊢ ((Rel ≼ ∧ 𝐴 ≼ ω) → 𝐴 ∈ V) | |
5 | 3, 4 | mpancom 687 | . . . . 5 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V) |
6 | 0sdomg 8630 | . . . . 5 ⊢ (𝐴 ∈ V → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ (𝐴 ≼ ω → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) |
8 | 7 | adantr 484 | . . 3 ⊢ ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) |
9 | 1, 8 | mpbird 260 | . 2 ⊢ ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → ∅ ≺ 𝐴) |
10 | nnenom 13343 | . . . . . 6 ⊢ ℕ ≈ ω | |
11 | 10 | ensymi 8542 | . . . . 5 ⊢ ω ≈ ℕ |
12 | 11 | a1i 11 | . . . 4 ⊢ (𝐴 ≼ ω → ω ≈ ℕ) |
13 | domentr 8551 | . . . 4 ⊢ ((𝐴 ≼ ω ∧ ω ≈ ℕ) → 𝐴 ≼ ℕ) | |
14 | 12, 13 | mpdan 686 | . . 3 ⊢ (𝐴 ≼ ω → 𝐴 ≼ ℕ) |
15 | 14 | adantr 484 | . 2 ⊢ ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → 𝐴 ≼ ℕ) |
16 | fodomr 8652 | . 2 ⊢ ((∅ ≺ 𝐴 ∧ 𝐴 ≼ ℕ) → ∃𝑓 𝑓:ℕ–onto→𝐴) | |
17 | 9, 15, 16 | syl2anc 587 | 1 ⊢ ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → ∃𝑓 𝑓:ℕ–onto→𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∃wex 1781 ∈ wcel 2111 ≠ wne 2987 Vcvv 3441 ∅c0 4243 class class class wbr 5030 Rel wrel 5524 –onto→wfo 6322 ωcom 7560 ≈ cen 8489 ≼ cdom 8490 ≺ csdm 8491 ℕcn 11625 |
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-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
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-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 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-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 |
This theorem is referenced by: ssnnf1octb 41822 issalnnd 42985 nnfoctbdj 43095 |
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