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 476 | . . 3 ⊢ ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → 𝐴 ≠ ∅) | |
2 | reldom 7847 | . . . . . . 7 ⊢ Rel ≼ | |
3 | 2 | a1i 11 | . . . . . 6 ⊢ (𝐴 ≼ ω → Rel ≼ ) |
4 | brrelex 5080 | . . . . . 6 ⊢ ((Rel ≼ ∧ 𝐴 ≼ ω) → 𝐴 ∈ V) | |
5 | 3, 4 | mpancom 700 | . . . . 5 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V) |
6 | 0sdomg 7974 | . . . . 5 ⊢ (𝐴 ∈ V → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ (𝐴 ≼ ω → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) |
8 | 7 | adantr 480 | . . 3 ⊢ ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) |
9 | 1, 8 | mpbird 246 | . 2 ⊢ ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → ∅ ≺ 𝐴) |
10 | nnenom 12641 | . . . . . 6 ⊢ ℕ ≈ ω | |
11 | 10 | ensymi 7892 | . . . . 5 ⊢ ω ≈ ℕ |
12 | 11 | a1i 11 | . . . 4 ⊢ (𝐴 ≼ ω → ω ≈ ℕ) |
13 | domentr 7901 | . . . 4 ⊢ ((𝐴 ≼ ω ∧ ω ≈ ℕ) → 𝐴 ≼ ℕ) | |
14 | 12, 13 | mpdan 699 | . . 3 ⊢ (𝐴 ≼ ω → 𝐴 ≼ ℕ) |
15 | 14 | adantr 480 | . 2 ⊢ ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → 𝐴 ≼ ℕ) |
16 | fodomr 7996 | . 2 ⊢ ((∅ ≺ 𝐴 ∧ 𝐴 ≼ ℕ) → ∃𝑓 𝑓:ℕ–onto→𝐴) | |
17 | 9, 15, 16 | syl2anc 691 | 1 ⊢ ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → ∃𝑓 𝑓:ℕ–onto→𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∃wex 1695 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 ∅c0 3874 class class class wbr 4583 Rel wrel 5043 –onto→wfo 5802 ωcom 6957 ≈ cen 7838 ≼ cdom 7839 ≺ csdm 7840 ℕcn 10897 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 |
This theorem is referenced by: ssnnf1octb 38377 issalnnd 39239 nnfoctbdj 39349 |
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