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Mirrors > Home > MPE Home > Th. List > gchaclem | Structured version Visualization version GIF version |
Description: Lemma for gchac 10105 (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 497 | . . 3 ⊢ (𝜑 → 𝐴 ≼ 𝐶) |
3 | reldom 8517 | . . . . . 6 ⊢ Rel ≼ | |
4 | 3 | brrelex2i 5611 | . . . . 5 ⊢ (𝐴 ≼ 𝐶 → 𝐶 ∈ V) |
5 | 2, 4 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ V) |
6 | canth2g 8673 | . . . 4 ⊢ (𝐶 ∈ V → 𝐶 ≺ 𝒫 𝐶) | |
7 | sdomdom 8539 | . . . 4 ⊢ (𝐶 ≺ 𝒫 𝐶 → 𝐶 ≼ 𝒫 𝐶) | |
8 | 5, 6, 7 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝐶 ≼ 𝒫 𝐶) |
9 | domtr 8564 | . . 3 ⊢ ((𝐴 ≼ 𝐶 ∧ 𝐶 ≼ 𝒫 𝐶) → 𝐴 ≼ 𝒫 𝐶) | |
10 | 2, 8, 9 | syl2anc 586 | . 2 ⊢ (𝜑 → 𝐴 ≼ 𝒫 𝐶) |
11 | gchaclem.3 | . . . . . 6 ⊢ (𝜑 → 𝒫 𝐶 ∈ GCH) | |
12 | 11 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≼ 𝒫 𝒫 𝐶) → 𝒫 𝐶 ∈ GCH) |
13 | gchaclem.1 | . . . . . . . 8 ⊢ (𝜑 → ω ≼ 𝐴) | |
14 | domtr 8564 | . . . . . . . 8 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ≼ 𝐶) → ω ≼ 𝐶) | |
15 | 13, 2, 14 | syl2anc 586 | . . . . . . 7 ⊢ (𝜑 → ω ≼ 𝐶) |
16 | 15 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≼ 𝒫 𝒫 𝐶) → ω ≼ 𝐶) |
17 | pwdjuidm 9619 | . . . . . 6 ⊢ (ω ≼ 𝐶 → (𝒫 𝐶 ⊔ 𝒫 𝐶) ≈ 𝒫 𝐶) | |
18 | 16, 17 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≼ 𝒫 𝒫 𝐶) → (𝒫 𝐶 ⊔ 𝒫 𝐶) ≈ 𝒫 𝐶) |
19 | simpr 487 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≼ 𝒫 𝒫 𝐶) → 𝐵 ≼ 𝒫 𝒫 𝐶) | |
20 | gchdomtri 10053 | . . . . 5 ⊢ ((𝒫 𝐶 ∈ GCH ∧ (𝒫 𝐶 ⊔ 𝒫 𝐶) ≈ 𝒫 𝐶 ∧ 𝐵 ≼ 𝒫 𝒫 𝐶) → (𝒫 𝐶 ≼ 𝐵 ∨ 𝐵 ≼ 𝒫 𝐶)) | |
21 | 12, 18, 19, 20 | syl3anc 1367 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ≼ 𝒫 𝒫 𝐶) → (𝒫 𝐶 ≼ 𝐵 ∨ 𝐵 ≼ 𝒫 𝐶)) |
22 | 21 | ex 415 | . . 3 ⊢ (𝜑 → (𝐵 ≼ 𝒫 𝒫 𝐶 → (𝒫 𝐶 ≼ 𝐵 ∨ 𝐵 ≼ 𝒫 𝐶))) |
23 | pwdom 8671 | . . . . 5 ⊢ (𝐴 ≼ 𝐶 → 𝒫 𝐴 ≼ 𝒫 𝐶) | |
24 | domtr 8564 | . . . . . 6 ⊢ ((𝒫 𝐴 ≼ 𝒫 𝐶 ∧ 𝒫 𝐶 ≼ 𝐵) → 𝒫 𝐴 ≼ 𝐵) | |
25 | 24 | ex 415 | . . . . 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 498 | . . . 4 ⊢ (𝜑 → (𝐵 ≼ 𝒫 𝐶 → 𝒫 𝐴 ≼ 𝐵)) |
28 | 26, 27 | jaod 855 | . . 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 514 | 1 ⊢ (𝜑 → (𝐴 ≼ 𝒫 𝐶 ∧ (𝐵 ≼ 𝒫 𝒫 𝐶 → 𝒫 𝐴 ≼ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 ∈ wcel 2114 Vcvv 3496 𝒫 cpw 4541 class class class wbr 5068 ωcom 7582 ≈ cen 8508 ≼ cdom 8509 ≺ csdm 8510 ⊔ cdju 9329 GCHcgch 10044 |
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-reu 3147 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-int 4879 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-2o 8105 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-wdom 9025 df-dju 9332 df-card 9370 df-gch 10045 |
This theorem is referenced by: (None) |
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