Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ssnct | Structured version Visualization version GIF version |
Description: A set containing an uncountable set is itself uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
Ref | Expression |
---|---|
ssnct.1 | ⊢ (𝜑 → ¬ 𝐴 ≼ ω) |
ssnct.2 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
Ref | Expression |
---|---|
ssnct | ⊢ (𝜑 → ¬ 𝐵 ≼ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssnct.2 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
2 | ssct 8704 | . . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ≼ ω) → 𝐴 ≼ ω) | |
3 | 1, 2 | sylan 583 | . 2 ⊢ ((𝜑 ∧ 𝐵 ≼ ω) → 𝐴 ≼ ω) |
4 | ssnct.1 | . . 3 ⊢ (𝜑 → ¬ 𝐴 ≼ ω) | |
5 | 4 | adantr 484 | . 2 ⊢ ((𝜑 ∧ 𝐵 ≼ ω) → ¬ 𝐴 ≼ ω) |
6 | 3, 5 | pm2.65da 817 | 1 ⊢ (𝜑 → ¬ 𝐵 ≼ ω) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ⊆ wss 3853 class class class wbr 5039 ωcom 7622 ≼ cdom 8602 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-dom 8606 |
This theorem is referenced by: iocnct 42694 iccnct 42695 |
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