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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 8560 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴) | |
2 | endom 8538 | . . . 4 ⊢ (𝐵 ≈ 𝐴 → 𝐵 ≼ 𝐴) | |
3 | acndom 9479 | . . . 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 8538 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵) | |
6 | acndom 9479 | . . . 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 214 | . 2 ⊢ (𝐴 ≈ 𝐵 → (𝑥 ∈ AC 𝐴 ↔ 𝑥 ∈ AC 𝐵)) |
9 | 8 | eqrdv 2821 | 1 ⊢ (𝐴 ≈ 𝐵 → AC 𝐴 = AC 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 ≈ cen 8508 ≼ cdom 8509 AC wacn 9369 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-1o 8104 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-fin 8515 df-acn 9373 |
This theorem is referenced by: (None) |
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