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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 8998 | . . 3 ⊢ (𝑋 ≈ 𝑌 → 𝑌 ≈ 𝑋) | |
2 | endom 8974 | . . 3 ⊢ (𝑌 ≈ 𝑋 → 𝑌 ≼ 𝑋) | |
3 | acndom2 10048 | . . 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 8974 | . . 3 ⊢ (𝑋 ≈ 𝑌 → 𝑋 ≼ 𝑌) | |
6 | acndom2 10048 | . . 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 5141 ≈ cen 8935 ≼ cdom 8936 AC wacn 9932 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7971 df-2nd 7972 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-acn 9936 |
This theorem is referenced by: (None) |
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