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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 9676 | . . . . 5 ⊢ ω ∈ On | |
2 | onenon 9979 | . . . . 5 ⊢ (ω ∈ On → ω ∈ dom card) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ ω ∈ dom card |
4 | domtri2 10019 | . . . 4 ⊢ ((ω ∈ dom card ∧ 𝑆 ∈ dom card) → (ω ≼ 𝑆 ↔ ¬ 𝑆 ≺ ω)) | |
5 | 3, 4 | mpan 688 | . . 3 ⊢ (𝑆 ∈ dom card → (ω ≼ 𝑆 ↔ ¬ 𝑆 ≺ ω)) |
6 | isfinite 9682 | . . . 4 ⊢ (𝑆 ∈ Fin ↔ 𝑆 ≺ ω) | |
7 | 6 | notbii 319 | . . 3 ⊢ (¬ 𝑆 ∈ Fin ↔ ¬ 𝑆 ≺ ω) |
8 | 5, 7 | bitr4di 288 | . 2 ⊢ (𝑆 ∈ dom card → (ω ≼ 𝑆 ↔ ¬ 𝑆 ∈ Fin)) |
9 | 8 | biimpar 476 | 1 ⊢ ((𝑆 ∈ dom card ∧ ¬ 𝑆 ∈ Fin) → ω ≼ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∈ wcel 2098 class class class wbr 5149 dom cdm 5678 Oncon0 6371 ωcom 7871 ≼ cdom 8962 ≺ csdm 8963 Fincfn 8964 cardccrd 9965 |
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 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9671 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-card 9969 |
This theorem is referenced by: isnumbasgrplem3 42673 |
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