Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 8772 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴) | |
2 | endom 8750 | . . . 4 ⊢ (𝐵 ≈ 𝐴 → 𝐵 ≼ 𝐴) | |
3 | acndom 9808 | . . . 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 8750 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵) | |
6 | acndom 9808 | . . . 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 2738 | 1 ⊢ (𝐴 ≈ 𝐵 → AC 𝐴 = AC 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 class class class wbr 5079 ≈ cen 8713 ≼ cdom 8714 AC wacn 9697 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-er 8481 df-map 8600 df-en 8717 df-dom 8718 df-fin 8720 df-acn 9701 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |