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Mirrors > Home > MPE Home > Th. List > gchdju1 | Structured version Visualization version GIF version |
Description: An infinite GCH-set is idempotent under cardinal successor. (Contributed by Mario Carneiro, 18-May-2015.) |
Ref | Expression |
---|---|
gchdju1 | ⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → (𝐴 ⊔ 1o) ≈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1onn 8652 | . . . . . 6 ⊢ 1o ∈ ω | |
2 | 1 | a1i 11 | . . . . 5 ⊢ (¬ 𝐴 ∈ Fin → 1o ∈ ω) |
3 | djudoml 10193 | . . . . 5 ⊢ ((𝐴 ∈ GCH ∧ 1o ∈ ω) → 𝐴 ≼ (𝐴 ⊔ 1o)) | |
4 | 2, 3 | sylan2 592 | . . . 4 ⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → 𝐴 ≼ (𝐴 ⊔ 1o)) |
5 | simpr 484 | . . . . . 6 ⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → ¬ 𝐴 ∈ Fin) | |
6 | nnfi 9181 | . . . . . . . . 9 ⊢ (1o ∈ ω → 1o ∈ Fin) | |
7 | 1, 6 | mp1i 13 | . . . . . . . 8 ⊢ (¬ 𝐴 ∈ Fin → 1o ∈ Fin) |
8 | fidomtri2 10003 | . . . . . . . 8 ⊢ ((𝐴 ∈ GCH ∧ 1o ∈ Fin) → (𝐴 ≼ 1o ↔ ¬ 1o ≺ 𝐴)) | |
9 | 7, 8 | sylan2 592 | . . . . . . 7 ⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → (𝐴 ≼ 1o ↔ ¬ 1o ≺ 𝐴)) |
10 | 1, 6 | mp1i 13 | . . . . . . . 8 ⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → 1o ∈ Fin) |
11 | domfi 9206 | . . . . . . . . 9 ⊢ ((1o ∈ Fin ∧ 𝐴 ≼ 1o) → 𝐴 ∈ Fin) | |
12 | 11 | ex 412 | . . . . . . . 8 ⊢ (1o ∈ Fin → (𝐴 ≼ 1o → 𝐴 ∈ Fin)) |
13 | 10, 12 | syl 17 | . . . . . . 7 ⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → (𝐴 ≼ 1o → 𝐴 ∈ Fin)) |
14 | 9, 13 | sylbird 260 | . . . . . 6 ⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → (¬ 1o ≺ 𝐴 → 𝐴 ∈ Fin)) |
15 | 5, 14 | mt3d 148 | . . . . 5 ⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → 1o ≺ 𝐴) |
16 | canthp1 10663 | . . . . 5 ⊢ (1o ≺ 𝐴 → (𝐴 ⊔ 1o) ≺ 𝒫 𝐴) | |
17 | 15, 16 | syl 17 | . . . 4 ⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → (𝐴 ⊔ 1o) ≺ 𝒫 𝐴) |
18 | 4, 17 | jca 511 | . . 3 ⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → (𝐴 ≼ (𝐴 ⊔ 1o) ∧ (𝐴 ⊔ 1o) ≺ 𝒫 𝐴)) |
19 | gchen1 10634 | . . 3 ⊢ (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴 ≼ (𝐴 ⊔ 1o) ∧ (𝐴 ⊔ 1o) ≺ 𝒫 𝐴)) → 𝐴 ≈ (𝐴 ⊔ 1o)) | |
20 | 18, 19 | mpdan 686 | . 2 ⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → 𝐴 ≈ (𝐴 ⊔ 1o)) |
21 | 20 | ensymd 9015 | 1 ⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → (𝐴 ⊔ 1o) ≈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∈ wcel 2099 𝒫 cpw 4598 class class class wbr 5142 ωcom 7862 1oc1o 8471 ≈ cen 8950 ≼ cdom 8951 ≺ csdm 8952 Fincfn 8953 ⊔ cdju 9907 GCHcgch 10629 |
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 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-inf2 9650 |
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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-2o 8479 df-er 8716 df-map 8836 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-oi 9519 df-dju 9910 df-card 9948 df-gch 10630 |
This theorem is referenced by: gchinf 10666 gchdjuidm 10677 gchpwdom 10679 |
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