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Mirrors > Home > MPE Home > Th. List > gchaclem | Structured version Visualization version GIF version |
Description: Lemma for gchac 9503 (obsolete, used in Sierpiński's proof). (Contributed by Mario Carneiro, 15-May-2015.) |
Ref | Expression |
---|---|
gchaclem.1 | ⊢ (𝜑 → ω ≼ 𝐴) |
gchaclem.3 | ⊢ (𝜑 → 𝒫 𝐶 ∈ GCH) |
gchaclem.4 | ⊢ (𝜑 → (𝐴 ≼ 𝐶 ∧ (𝐵 ≼ 𝒫 𝐶 → 𝒫 𝐴 ≼ 𝐵))) |
Ref | Expression |
---|---|
gchaclem | ⊢ (𝜑 → (𝐴 ≼ 𝒫 𝐶 ∧ (𝐵 ≼ 𝒫 𝒫 𝐶 → 𝒫 𝐴 ≼ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gchaclem.4 | . . . 4 ⊢ (𝜑 → (𝐴 ≼ 𝐶 ∧ (𝐵 ≼ 𝒫 𝐶 → 𝒫 𝐴 ≼ 𝐵))) | |
2 | 1 | simpld 475 | . . 3 ⊢ (𝜑 → 𝐴 ≼ 𝐶) |
3 | reldom 7961 | . . . . . 6 ⊢ Rel ≼ | |
4 | 3 | brrelex2i 5159 | . . . . 5 ⊢ (𝐴 ≼ 𝐶 → 𝐶 ∈ V) |
5 | 2, 4 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ V) |
6 | canth2g 8114 | . . . 4 ⊢ (𝐶 ∈ V → 𝐶 ≺ 𝒫 𝐶) | |
7 | sdomdom 7983 | . . . 4 ⊢ (𝐶 ≺ 𝒫 𝐶 → 𝐶 ≼ 𝒫 𝐶) | |
8 | 5, 6, 7 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝐶 ≼ 𝒫 𝐶) |
9 | domtr 8009 | . . 3 ⊢ ((𝐴 ≼ 𝐶 ∧ 𝐶 ≼ 𝒫 𝐶) → 𝐴 ≼ 𝒫 𝐶) | |
10 | 2, 8, 9 | syl2anc 693 | . 2 ⊢ (𝜑 → 𝐴 ≼ 𝒫 𝐶) |
11 | gchaclem.3 | . . . . . 6 ⊢ (𝜑 → 𝒫 𝐶 ∈ GCH) | |
12 | 11 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≼ 𝒫 𝒫 𝐶) → 𝒫 𝐶 ∈ GCH) |
13 | gchaclem.1 | . . . . . . . 8 ⊢ (𝜑 → ω ≼ 𝐴) | |
14 | domtr 8009 | . . . . . . . 8 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ≼ 𝐶) → ω ≼ 𝐶) | |
15 | 13, 2, 14 | syl2anc 693 | . . . . . . 7 ⊢ (𝜑 → ω ≼ 𝐶) |
16 | 15 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≼ 𝒫 𝒫 𝐶) → ω ≼ 𝐶) |
17 | pwcdaidm 9017 | . . . . . 6 ⊢ (ω ≼ 𝐶 → (𝒫 𝐶 +𝑐 𝒫 𝐶) ≈ 𝒫 𝐶) | |
18 | 16, 17 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≼ 𝒫 𝒫 𝐶) → (𝒫 𝐶 +𝑐 𝒫 𝐶) ≈ 𝒫 𝐶) |
19 | simpr 477 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≼ 𝒫 𝒫 𝐶) → 𝐵 ≼ 𝒫 𝒫 𝐶) | |
20 | gchdomtri 9451 | . . . . 5 ⊢ ((𝒫 𝐶 ∈ GCH ∧ (𝒫 𝐶 +𝑐 𝒫 𝐶) ≈ 𝒫 𝐶 ∧ 𝐵 ≼ 𝒫 𝒫 𝐶) → (𝒫 𝐶 ≼ 𝐵 ∨ 𝐵 ≼ 𝒫 𝐶)) | |
21 | 12, 18, 19, 20 | syl3anc 1326 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ≼ 𝒫 𝒫 𝐶) → (𝒫 𝐶 ≼ 𝐵 ∨ 𝐵 ≼ 𝒫 𝐶)) |
22 | 21 | ex 450 | . . 3 ⊢ (𝜑 → (𝐵 ≼ 𝒫 𝒫 𝐶 → (𝒫 𝐶 ≼ 𝐵 ∨ 𝐵 ≼ 𝒫 𝐶))) |
23 | pwdom 8112 | . . . . 5 ⊢ (𝐴 ≼ 𝐶 → 𝒫 𝐴 ≼ 𝒫 𝐶) | |
24 | domtr 8009 | . . . . . 6 ⊢ ((𝒫 𝐴 ≼ 𝒫 𝐶 ∧ 𝒫 𝐶 ≼ 𝐵) → 𝒫 𝐴 ≼ 𝐵) | |
25 | 24 | ex 450 | . . . . 5 ⊢ (𝒫 𝐴 ≼ 𝒫 𝐶 → (𝒫 𝐶 ≼ 𝐵 → 𝒫 𝐴 ≼ 𝐵)) |
26 | 2, 23, 25 | 3syl 18 | . . . 4 ⊢ (𝜑 → (𝒫 𝐶 ≼ 𝐵 → 𝒫 𝐴 ≼ 𝐵)) |
27 | 1 | simprd 479 | . . . 4 ⊢ (𝜑 → (𝐵 ≼ 𝒫 𝐶 → 𝒫 𝐴 ≼ 𝐵)) |
28 | 26, 27 | jaod 395 | . . 3 ⊢ (𝜑 → ((𝒫 𝐶 ≼ 𝐵 ∨ 𝐵 ≼ 𝒫 𝐶) → 𝒫 𝐴 ≼ 𝐵)) |
29 | 22, 28 | syld 47 | . 2 ⊢ (𝜑 → (𝐵 ≼ 𝒫 𝒫 𝐶 → 𝒫 𝐴 ≼ 𝐵)) |
30 | 10, 29 | jca 554 | 1 ⊢ (𝜑 → (𝐴 ≼ 𝒫 𝐶 ∧ (𝐵 ≼ 𝒫 𝒫 𝐶 → 𝒫 𝐴 ≼ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 383 ∧ wa 384 ∈ wcel 1990 Vcvv 3200 𝒫 cpw 4158 class class class wbr 4653 (class class class)co 6650 ωcom 7065 ≈ cen 7952 ≼ cdom 7953 ≺ csdm 7954 +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-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 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-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-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-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-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-1o 7560 df-2o 7561 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-wdom 8464 df-card 8765 df-cda 8990 df-gch 9443 |
This theorem is referenced by: (None) |
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