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Mirrors > Home > MPE Home > Th. List > gchcda1 | 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 |
---|---|
gchcda1 | ⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → (𝐴 +𝑐 1𝑜) ≈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1onn 7719 | . . . . . 6 ⊢ 1𝑜 ∈ ω | |
2 | 1 | a1i 11 | . . . . 5 ⊢ (¬ 𝐴 ∈ Fin → 1𝑜 ∈ ω) |
3 | cdadom3 9010 | . . . . 5 ⊢ ((𝐴 ∈ GCH ∧ 1𝑜 ∈ ω) → 𝐴 ≼ (𝐴 +𝑐 1𝑜)) | |
4 | 2, 3 | sylan2 491 | . . . 4 ⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → 𝐴 ≼ (𝐴 +𝑐 1𝑜)) |
5 | simpr 477 | . . . . . 6 ⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → ¬ 𝐴 ∈ Fin) | |
6 | nnfi 8153 | . . . . . . . . 9 ⊢ (1𝑜 ∈ ω → 1𝑜 ∈ Fin) | |
7 | 1, 6 | mp1i 13 | . . . . . . . 8 ⊢ (¬ 𝐴 ∈ Fin → 1𝑜 ∈ Fin) |
8 | fidomtri2 8820 | . . . . . . . 8 ⊢ ((𝐴 ∈ GCH ∧ 1𝑜 ∈ Fin) → (𝐴 ≼ 1𝑜 ↔ ¬ 1𝑜 ≺ 𝐴)) | |
9 | 7, 8 | sylan2 491 | . . . . . . 7 ⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → (𝐴 ≼ 1𝑜 ↔ ¬ 1𝑜 ≺ 𝐴)) |
10 | 1, 6 | mp1i 13 | . . . . . . . 8 ⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → 1𝑜 ∈ Fin) |
11 | domfi 8181 | . . . . . . . . 9 ⊢ ((1𝑜 ∈ Fin ∧ 𝐴 ≼ 1𝑜) → 𝐴 ∈ Fin) | |
12 | 11 | ex 450 | . . . . . . . 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 250 | . . . . . 6 ⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → (¬ 1𝑜 ≺ 𝐴 → 𝐴 ∈ Fin)) |
15 | 5, 14 | mt3d 140 | . . . . 5 ⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → 1𝑜 ≺ 𝐴) |
16 | canthp1 9476 | . . . . 5 ⊢ (1𝑜 ≺ 𝐴 → (𝐴 +𝑐 1𝑜) ≺ 𝒫 𝐴) | |
17 | 15, 16 | syl 17 | . . . 4 ⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → (𝐴 +𝑐 1𝑜) ≺ 𝒫 𝐴) |
18 | 4, 17 | jca 554 | . . 3 ⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → (𝐴 ≼ (𝐴 +𝑐 1𝑜) ∧ (𝐴 +𝑐 1𝑜) ≺ 𝒫 𝐴)) |
19 | gchen1 9447 | . . 3 ⊢ (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴 ≼ (𝐴 +𝑐 1𝑜) ∧ (𝐴 +𝑐 1𝑜) ≺ 𝒫 𝐴)) → 𝐴 ≈ (𝐴 +𝑐 1𝑜)) | |
20 | 18, 19 | mpdan 702 | . 2 ⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → 𝐴 ≈ (𝐴 +𝑐 1𝑜)) |
21 | 20 | ensymd 8007 | 1 ⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → (𝐴 +𝑐 1𝑜) ≈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 ∈ wcel 1990 𝒫 cpw 4158 class class class wbr 4653 (class class class)co 6650 ωcom 7065 1𝑜c1o 7553 ≈ cen 7952 ≼ cdom 7953 ≺ csdm 7954 Fincfn 7955 +𝑐 ccda 8989 GCHcgch 9442 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-oi 8415 df-card 8765 df-cda 8990 df-gch 9443 |
This theorem is referenced by: gchinf 9479 gchcdaidm 9490 gchpwdom 9492 |
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