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Mirrors > Home > MPE Home > Th. List > gchaclem | Structured version Visualization version GIF version |
Description: Lemma for gchac 10750 (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 9009 | . . . . . 6 ⊢ Rel ≼ | |
4 | 3 | brrelex2i 5757 | . . . . 5 ⊢ (𝐴 ≼ 𝐶 → 𝐶 ∈ V) |
5 | 2, 4 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ V) |
6 | canth2g 9197 | . . . 4 ⊢ (𝐶 ∈ V → 𝐶 ≺ 𝒫 𝐶) | |
7 | sdomdom 9040 | . . . 4 ⊢ (𝐶 ≺ 𝒫 𝐶 → 𝐶 ≼ 𝒫 𝐶) | |
8 | 5, 6, 7 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝐶 ≼ 𝒫 𝐶) |
9 | domtr 9067 | . . 3 ⊢ ((𝐴 ≼ 𝐶 ∧ 𝐶 ≼ 𝒫 𝐶) → 𝐴 ≼ 𝒫 𝐶) | |
10 | 2, 8, 9 | syl2anc 583 | . 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 9067 | . . . . . . . 8 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ≼ 𝐶) → ω ≼ 𝐶) | |
15 | 13, 2, 14 | syl2anc 583 | . . . . . . 7 ⊢ (𝜑 → ω ≼ 𝐶) |
16 | 15 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≼ 𝒫 𝒫 𝐶) → ω ≼ 𝐶) |
17 | pwdjuidm 10261 | . . . . . 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 10698 | . . . . 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 1371 | . . . 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 9195 | . . . . 5 ⊢ (𝐴 ≼ 𝐶 → 𝒫 𝐴 ≼ 𝒫 𝐶) | |
24 | domtr 9067 | . . . . . 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 858 | . . 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 846 ∈ wcel 2108 Vcvv 3488 𝒫 cpw 4622 class class class wbr 5166 ωcom 7903 ≈ cen 9000 ≼ cdom 9001 ≺ csdm 9002 ⊔ cdju 9967 GCHcgch 10689 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-wdom 9634 df-dju 9970 df-card 10008 df-gch 10690 |
This theorem is referenced by: (None) |
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