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Mirrors > Home > MPE Home > Th. List > gchaclem | Structured version Visualization version GIF version |
Description: Lemma for gchac 9899 (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 487 | . . 3 ⊢ (𝜑 → 𝐴 ≼ 𝐶) |
3 | reldom 8310 | . . . . . 6 ⊢ Rel ≼ | |
4 | 3 | brrelex2i 5455 | . . . . 5 ⊢ (𝐴 ≼ 𝐶 → 𝐶 ∈ V) |
5 | 2, 4 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ V) |
6 | canth2g 8465 | . . . 4 ⊢ (𝐶 ∈ V → 𝐶 ≺ 𝒫 𝐶) | |
7 | sdomdom 8332 | . . . 4 ⊢ (𝐶 ≺ 𝒫 𝐶 → 𝐶 ≼ 𝒫 𝐶) | |
8 | 5, 6, 7 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝐶 ≼ 𝒫 𝐶) |
9 | domtr 8357 | . . 3 ⊢ ((𝐴 ≼ 𝐶 ∧ 𝐶 ≼ 𝒫 𝐶) → 𝐴 ≼ 𝒫 𝐶) | |
10 | 2, 8, 9 | syl2anc 576 | . 2 ⊢ (𝜑 → 𝐴 ≼ 𝒫 𝐶) |
11 | gchaclem.3 | . . . . . 6 ⊢ (𝜑 → 𝒫 𝐶 ∈ GCH) | |
12 | 11 | adantr 473 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≼ 𝒫 𝒫 𝐶) → 𝒫 𝐶 ∈ GCH) |
13 | gchaclem.1 | . . . . . . . 8 ⊢ (𝜑 → ω ≼ 𝐴) | |
14 | domtr 8357 | . . . . . . . 8 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ≼ 𝐶) → ω ≼ 𝐶) | |
15 | 13, 2, 14 | syl2anc 576 | . . . . . . 7 ⊢ (𝜑 → ω ≼ 𝐶) |
16 | 15 | adantr 473 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≼ 𝒫 𝒫 𝐶) → ω ≼ 𝐶) |
17 | pwdjuidm 9413 | . . . . . 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 477 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≼ 𝒫 𝒫 𝐶) → 𝐵 ≼ 𝒫 𝒫 𝐶) | |
20 | gchdomtri 9847 | . . . . 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 1351 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ≼ 𝒫 𝒫 𝐶) → (𝒫 𝐶 ≼ 𝐵 ∨ 𝐵 ≼ 𝒫 𝐶)) |
22 | 21 | ex 405 | . . 3 ⊢ (𝜑 → (𝐵 ≼ 𝒫 𝒫 𝐶 → (𝒫 𝐶 ≼ 𝐵 ∨ 𝐵 ≼ 𝒫 𝐶))) |
23 | pwdom 8463 | . . . . 5 ⊢ (𝐴 ≼ 𝐶 → 𝒫 𝐴 ≼ 𝒫 𝐶) | |
24 | domtr 8357 | . . . . . 6 ⊢ ((𝒫 𝐴 ≼ 𝒫 𝐶 ∧ 𝒫 𝐶 ≼ 𝐵) → 𝒫 𝐴 ≼ 𝐵) | |
25 | 24 | ex 405 | . . . . 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 488 | . . . 4 ⊢ (𝜑 → (𝐵 ≼ 𝒫 𝐶 → 𝒫 𝐴 ≼ 𝐵)) |
28 | 26, 27 | jaod 845 | . . 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 504 | 1 ⊢ (𝜑 → (𝐴 ≼ 𝒫 𝐶 ∧ (𝐵 ≼ 𝒫 𝒫 𝐶 → 𝒫 𝐴 ≼ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∨ wo 833 ∈ wcel 2050 Vcvv 3409 𝒫 cpw 4416 class class class wbr 4925 ωcom 7394 ≈ cen 8301 ≼ cdom 8302 ≺ csdm 8303 ⊔ cdju 9119 GCHcgch 9838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7499 df-2nd 7500 df-1o 7903 df-2o 7904 df-er 8087 df-map 8206 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-wdom 8816 df-dju 9122 df-card 9160 df-gch 9839 |
This theorem is referenced by: (None) |
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