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Mirrors > Home > MPE Home > Th. List > acnen | Structured version Visualization version GIF version |
Description: The class of choice sets of length 𝐴 is a cardinal invariant. (Contributed by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
acnen | ⊢ (𝐴 ≈ 𝐵 → AC 𝐴 = AC 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ensym 9024 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴) | |
2 | endom 9000 | . . . 4 ⊢ (𝐵 ≈ 𝐴 → 𝐵 ≼ 𝐴) | |
3 | acndom 10075 | . . . 4 ⊢ (𝐵 ≼ 𝐴 → (𝑥 ∈ AC 𝐴 → 𝑥 ∈ AC 𝐵)) | |
4 | 1, 2, 3 | 3syl 18 | . . 3 ⊢ (𝐴 ≈ 𝐵 → (𝑥 ∈ AC 𝐴 → 𝑥 ∈ AC 𝐵)) |
5 | endom 9000 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵) | |
6 | acndom 10075 | . . . 4 ⊢ (𝐴 ≼ 𝐵 → (𝑥 ∈ AC 𝐵 → 𝑥 ∈ AC 𝐴)) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝐴 ≈ 𝐵 → (𝑥 ∈ AC 𝐵 → 𝑥 ∈ AC 𝐴)) |
8 | 4, 7 | impbid 211 | . 2 ⊢ (𝐴 ≈ 𝐵 → (𝑥 ∈ AC 𝐴 ↔ 𝑥 ∈ AC 𝐵)) |
9 | 8 | eqrdv 2726 | 1 ⊢ (𝐴 ≈ 𝐵 → AC 𝐴 = AC 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 class class class wbr 5148 ≈ cen 8961 ≼ cdom 8962 AC wacn 9962 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-fin 8968 df-acn 9966 |
This theorem is referenced by: (None) |
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