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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 9039 | . . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ≼ ω) → 𝐴 ≼ ω) | |
3 | 1, 2 | sylan 581 | . 2 ⊢ ((𝜑 ∧ 𝐵 ≼ ω) → 𝐴 ≼ ω) |
4 | ssnct.1 | . . 3 ⊢ (𝜑 → ¬ 𝐴 ≼ ω) | |
5 | 4 | adantr 482 | . 2 ⊢ ((𝜑 ∧ 𝐵 ≼ ω) → ¬ 𝐴 ≼ ω) |
6 | 3, 5 | pm2.65da 816 | 1 ⊢ (𝜑 → ¬ 𝐵 ≼ ω) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ⊆ wss 3946 class class class wbr 5144 ωcom 7842 ≼ cdom 8925 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5145 df-opab 5207 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-dom 8929 |
This theorem is referenced by: iocnct 44126 iccnct 44127 |
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