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Mirrors > Home > MPE Home > Th. List > Mathboxes > numinfctb | Structured version Visualization version GIF version |
Description: A numerable infinite set contains a countable subset. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.) |
Ref | Expression |
---|---|
numinfctb | ⊢ ((𝑆 ∈ dom card ∧ ¬ 𝑆 ∈ Fin) → ω ≼ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omelon 9404 | . . . . 5 ⊢ ω ∈ On | |
2 | onenon 9707 | . . . . 5 ⊢ (ω ∈ On → ω ∈ dom card) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ ω ∈ dom card |
4 | domtri2 9747 | . . . 4 ⊢ ((ω ∈ dom card ∧ 𝑆 ∈ dom card) → (ω ≼ 𝑆 ↔ ¬ 𝑆 ≺ ω)) | |
5 | 3, 4 | mpan 687 | . . 3 ⊢ (𝑆 ∈ dom card → (ω ≼ 𝑆 ↔ ¬ 𝑆 ≺ ω)) |
6 | isfinite 9410 | . . . 4 ⊢ (𝑆 ∈ Fin ↔ 𝑆 ≺ ω) | |
7 | 6 | notbii 320 | . . 3 ⊢ (¬ 𝑆 ∈ Fin ↔ ¬ 𝑆 ≺ ω) |
8 | 5, 7 | bitr4di 289 | . 2 ⊢ (𝑆 ∈ dom card → (ω ≼ 𝑆 ↔ ¬ 𝑆 ∈ Fin)) |
9 | 8 | biimpar 478 | 1 ⊢ ((𝑆 ∈ dom card ∧ ¬ 𝑆 ∈ Fin) → ω ≼ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∈ wcel 2106 class class class wbr 5074 dom cdm 5589 Oncon0 6266 ωcom 7712 ≼ cdom 8731 ≺ csdm 8732 Fincfn 8733 cardccrd 9693 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-om 7713 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-card 9697 |
This theorem is referenced by: isnumbasgrplem3 40930 |
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