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Mirrors > Home > MPE Home > Th. List > acnen2 | Structured version Visualization version GIF version |
Description: The class of sets with choice sequences of length 𝐴 is a cardinal invariant. (Contributed by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
acnen2 | ⊢ (𝑋 ≈ 𝑌 → (𝑋 ∈ AC 𝐴 ↔ 𝑌 ∈ AC 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ensym 9028 | . . 3 ⊢ (𝑋 ≈ 𝑌 → 𝑌 ≈ 𝑋) | |
2 | endom 9004 | . . 3 ⊢ (𝑌 ≈ 𝑋 → 𝑌 ≼ 𝑋) | |
3 | acndom2 10083 | . . 3 ⊢ (𝑌 ≼ 𝑋 → (𝑋 ∈ AC 𝐴 → 𝑌 ∈ AC 𝐴)) | |
4 | 1, 2, 3 | 3syl 18 | . 2 ⊢ (𝑋 ≈ 𝑌 → (𝑋 ∈ AC 𝐴 → 𝑌 ∈ AC 𝐴)) |
5 | endom 9004 | . . 3 ⊢ (𝑋 ≈ 𝑌 → 𝑋 ≼ 𝑌) | |
6 | acndom2 10083 | . . 3 ⊢ (𝑋 ≼ 𝑌 → (𝑌 ∈ AC 𝐴 → 𝑋 ∈ AC 𝐴)) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (𝑋 ≈ 𝑌 → (𝑌 ∈ AC 𝐴 → 𝑋 ∈ AC 𝐴)) |
8 | 4, 7 | impbid 211 | 1 ⊢ (𝑋 ≈ 𝑌 → (𝑋 ∈ AC 𝐴 ↔ 𝑌 ∈ AC 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2098 class class class wbr 5150 ≈ cen 8965 ≼ cdom 8966 AC wacn 9967 |
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 2698 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 7997 df-2nd 7998 df-er 8729 df-map 8851 df-en 8969 df-dom 8970 df-acn 9971 |
This theorem is referenced by: (None) |
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