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 7926 | . . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ≼ ω) → 𝐴 ≼ ω) | |
3 | 1, 2 | sylan 487 | . 2 ⊢ ((𝜑 ∧ 𝐵 ≼ ω) → 𝐴 ≼ ω) |
4 | ssnct.1 | . . 3 ⊢ (𝜑 → ¬ 𝐴 ≼ ω) | |
5 | 4 | adantr 480 | . 2 ⊢ ((𝜑 ∧ 𝐵 ≼ ω) → ¬ 𝐴 ≼ ω) |
6 | 3, 5 | pm2.65da 598 | 1 ⊢ (𝜑 → ¬ 𝐵 ≼ ω) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ⊆ wss 3540 class class class wbr 4583 ωcom 6957 ≼ cdom 7839 |
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 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 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-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 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-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-dom 7843 |
This theorem is referenced by: iocnct 38614 iccnct 38615 |
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