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Mirrors > Home > MPE Home > Th. List > engch | Structured version Visualization version GIF version |
Description: The property of being a GCH-set is a cardinal invariant. (Contributed by Mario Carneiro, 15-May-2015.) |
Ref | Expression |
---|---|
engch | ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ GCH ↔ 𝐵 ∈ GCH)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | enfi 8061 | . . 3 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin)) | |
2 | sdomen1 7989 | . . . . . 6 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ≺ 𝑥 ↔ 𝐵 ≺ 𝑥)) | |
3 | pwen 8018 | . . . . . . 7 ⊢ (𝐴 ≈ 𝐵 → 𝒫 𝐴 ≈ 𝒫 𝐵) | |
4 | sdomen2 7990 | . . . . . . 7 ⊢ (𝒫 𝐴 ≈ 𝒫 𝐵 → (𝑥 ≺ 𝒫 𝐴 ↔ 𝑥 ≺ 𝒫 𝐵)) | |
5 | 3, 4 | syl 17 | . . . . . 6 ⊢ (𝐴 ≈ 𝐵 → (𝑥 ≺ 𝒫 𝐴 ↔ 𝑥 ≺ 𝒫 𝐵)) |
6 | 2, 5 | anbi12d 743 | . . . . 5 ⊢ (𝐴 ≈ 𝐵 → ((𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴) ↔ (𝐵 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐵))) |
7 | 6 | notbid 307 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → (¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴) ↔ ¬ (𝐵 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐵))) |
8 | 7 | albidv 1836 | . . 3 ⊢ (𝐴 ≈ 𝐵 → (∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴) ↔ ∀𝑥 ¬ (𝐵 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐵))) |
9 | 1, 8 | orbi12d 742 | . 2 ⊢ (𝐴 ≈ 𝐵 → ((𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴)) ↔ (𝐵 ∈ Fin ∨ ∀𝑥 ¬ (𝐵 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐵)))) |
10 | relen 7846 | . . . 4 ⊢ Rel ≈ | |
11 | 10 | brrelexi 5082 | . . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ∈ V) |
12 | elgch 9323 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ GCH ↔ (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴)))) | |
13 | 11, 12 | syl 17 | . 2 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ GCH ↔ (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴)))) |
14 | 10 | brrelex2i 5083 | . . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ∈ V) |
15 | elgch 9323 | . . 3 ⊢ (𝐵 ∈ V → (𝐵 ∈ GCH ↔ (𝐵 ∈ Fin ∨ ∀𝑥 ¬ (𝐵 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐵)))) | |
16 | 14, 15 | syl 17 | . 2 ⊢ (𝐴 ≈ 𝐵 → (𝐵 ∈ GCH ↔ (𝐵 ∈ Fin ∨ ∀𝑥 ¬ (𝐵 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐵)))) |
17 | 9, 13, 16 | 3bitr4d 299 | 1 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ GCH ↔ 𝐵 ∈ GCH)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 ∀wal 1473 ∈ wcel 1977 Vcvv 3173 𝒫 cpw 4108 class class class wbr 4583 ≈ cen 7838 ≺ csdm 7840 Fincfn 7841 GCHcgch 9321 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-1o 7447 df-2o 7448 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-gch 9322 |
This theorem is referenced by: gch2 9376 |
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