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Mirrors > Home > MPE Home > Th. List > gchaclem | Structured version Visualization version GIF version |
Description: Lemma for gchac 10719 (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 494 | . . 3 ⊢ (𝜑 → 𝐴 ≼ 𝐶) |
3 | reldom 8990 | . . . . . 6 ⊢ Rel ≼ | |
4 | 3 | brrelex2i 5746 | . . . . 5 ⊢ (𝐴 ≼ 𝐶 → 𝐶 ∈ V) |
5 | 2, 4 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ V) |
6 | canth2g 9170 | . . . 4 ⊢ (𝐶 ∈ V → 𝐶 ≺ 𝒫 𝐶) | |
7 | sdomdom 9019 | . . . 4 ⊢ (𝐶 ≺ 𝒫 𝐶 → 𝐶 ≼ 𝒫 𝐶) | |
8 | 5, 6, 7 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝐶 ≼ 𝒫 𝐶) |
9 | domtr 9046 | . . 3 ⊢ ((𝐴 ≼ 𝐶 ∧ 𝐶 ≼ 𝒫 𝐶) → 𝐴 ≼ 𝒫 𝐶) | |
10 | 2, 8, 9 | syl2anc 584 | . 2 ⊢ (𝜑 → 𝐴 ≼ 𝒫 𝐶) |
11 | gchaclem.3 | . . . . . 6 ⊢ (𝜑 → 𝒫 𝐶 ∈ GCH) | |
12 | 11 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≼ 𝒫 𝒫 𝐶) → 𝒫 𝐶 ∈ GCH) |
13 | gchaclem.1 | . . . . . . . 8 ⊢ (𝜑 → ω ≼ 𝐴) | |
14 | domtr 9046 | . . . . . . . 8 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ≼ 𝐶) → ω ≼ 𝐶) | |
15 | 13, 2, 14 | syl2anc 584 | . . . . . . 7 ⊢ (𝜑 → ω ≼ 𝐶) |
16 | 15 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≼ 𝒫 𝒫 𝐶) → ω ≼ 𝐶) |
17 | pwdjuidm 10230 | . . . . . 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 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≼ 𝒫 𝒫 𝐶) → 𝐵 ≼ 𝒫 𝒫 𝐶) | |
20 | gchdomtri 10667 | . . . . 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 1370 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ≼ 𝒫 𝒫 𝐶) → (𝒫 𝐶 ≼ 𝐵 ∨ 𝐵 ≼ 𝒫 𝐶)) |
22 | 21 | ex 412 | . . 3 ⊢ (𝜑 → (𝐵 ≼ 𝒫 𝒫 𝐶 → (𝒫 𝐶 ≼ 𝐵 ∨ 𝐵 ≼ 𝒫 𝐶))) |
23 | pwdom 9168 | . . . . 5 ⊢ (𝐴 ≼ 𝐶 → 𝒫 𝐴 ≼ 𝒫 𝐶) | |
24 | domtr 9046 | . . . . . 6 ⊢ ((𝒫 𝐴 ≼ 𝒫 𝐶 ∧ 𝒫 𝐶 ≼ 𝐵) → 𝒫 𝐴 ≼ 𝐵) | |
25 | 24 | ex 412 | . . . . 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 495 | . . . 4 ⊢ (𝜑 → (𝐵 ≼ 𝒫 𝐶 → 𝒫 𝐴 ≼ 𝐵)) |
28 | 26, 27 | jaod 859 | . . 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 511 | 1 ⊢ (𝜑 → (𝐴 ≼ 𝒫 𝐶 ∧ (𝐵 ≼ 𝒫 𝒫 𝐶 → 𝒫 𝐴 ≼ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 ∈ wcel 2106 Vcvv 3478 𝒫 cpw 4605 class class class wbr 5148 ωcom 7887 ≈ cen 8981 ≼ cdom 8982 ≺ csdm 8983 ⊔ cdju 9936 GCHcgch 10658 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-wdom 9603 df-dju 9939 df-card 9977 df-gch 10659 |
This theorem is referenced by: (None) |
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