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Theorem gchaclem 10599

Description: Lemma for gchac 10602 (obsolete, used in Sierpiński's proof). (Contributed by Mario Carneiro, 15-May-2015.)

**Hypotheses**
Ref	Expression
gchaclem.1	⊢ (𝜑 → ω ≼ 𝐴)
gchaclem.3	⊢ (𝜑 → 𝒫 𝐶 ∈ GCH)
gchaclem.4	⊢ (𝜑 → (𝐴 ≼ 𝐶 ∧ (𝐵 ≼ 𝒫 𝐶 → 𝒫 𝐴 ≼ 𝐵)))

**Assertion**
Ref	Expression
gchaclem	⊢ (𝜑 → (𝐴 ≼ 𝒫 𝐶 ∧ (𝐵 ≼ 𝒫 𝒫 𝐶 → 𝒫 𝐴 ≼ 𝐵)))

**Proof of Theorem gchaclem**
Step	Hyp	Ref	Expression
1		gchaclem.4	. . . 4 ⊢ (𝜑 → (𝐴 ≼ 𝐶 ∧ (𝐵 ≼ 𝒫 𝐶 → 𝒫 𝐴 ≼ 𝐵)))
2	1	simpld 495	. . 3 ⊢ (𝜑 → 𝐴 ≼ 𝐶)
3		reldom 8896	. . . . . 6 ⊢ Rel ≼
4	3	brrelex2i 5682	. . . . 5 ⊢ (𝐴 ≼ 𝐶 → 𝐶 ∈ V)
5	2, 4	syl 17	. . . 4 ⊢ (𝜑 → 𝐶 ∈ V)
6		canth2g 9066	. . . 4 ⊢ (𝐶 ∈ V → 𝐶 ≺ 𝒫 𝐶)
7		sdomdom 8924	. . . 4 ⊢ (𝐶 ≺ 𝒫 𝐶 → 𝐶 ≼ 𝒫 𝐶)
8	5, 6, 7	3syl 18	. . 3 ⊢ (𝜑 → 𝐶 ≼ 𝒫 𝐶)
9		domtr 8951	. . 3 ⊢ ((𝐴 ≼ 𝐶 ∧ 𝐶 ≼ 𝒫 𝐶) → 𝐴 ≼ 𝒫 𝐶)
10	2, 8, 9	syl2anc 590	. 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 8951	. . . . . . . 8 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ≼ 𝐶) → ω ≼ 𝐶)
15	13, 2, 14	syl2anc 590	. . . . . . 7 ⊢ (𝜑 → ω ≼ 𝐶)
16	15	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≼ 𝒫 𝒫 𝐶) → ω ≼ 𝐶)
17		pwdjuidm 10112	. . . . . 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 485	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≼ 𝒫 𝒫 𝐶) → 𝐵 ≼ 𝒫 𝒫 𝐶)
20		gchdomtri 10550	. . . . 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 1379	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≼ 𝒫 𝒫 𝐶) → (𝒫 𝐶 ≼ 𝐵 ∨ 𝐵 ≼ 𝒫 𝐶))
22	21	ex 413	. . 3 ⊢ (𝜑 → (𝐵 ≼ 𝒫 𝒫 𝐶 → (𝒫 𝐶 ≼ 𝐵 ∨ 𝐵 ≼ 𝒫 𝐶)))
23		pwdom 9064	. . . . 5 ⊢ (𝐴 ≼ 𝐶 → 𝒫 𝐴 ≼ 𝒫 𝐶)
24		domtr 8951	. . . . . 6 ⊢ ((𝒫 𝐴 ≼ 𝒫 𝐶 ∧ 𝒫 𝐶 ≼ 𝐵) → 𝒫 𝐴 ≼ 𝐵)
25	24	ex 413	. . . . 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 496	. . . 4 ⊢ (𝜑 → (𝐵 ≼ 𝒫 𝐶 → 𝒫 𝐴 ≼ 𝐵))
28	26, 27	jaod 865	. . 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 516	1 ⊢ (𝜑 → (𝐴 ≼ 𝒫 𝐶 ∧ (𝐵 ≼ 𝒫 𝒫 𝐶 → 𝒫 𝐴 ≼ 𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∨ wo 853 ∈ wcel 2119 Vcvv 3432 𝒫 cpw 4536 class class class wbr 5079 ωcom 7813 ≈ cen 8887 ≼ cdom 8888 ≺ csdm 8889 ⊔ cdju 9820 GCHcgch 10541

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-ral 3055 df-rex 3065 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-ov 7366 df-oprab 7367 df-mpo 7368 df-om 7814 df-1st 7938 df-2nd 7939 df-1o 8402 df-2o 8403 df-er 8640 df-map 8772 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-wdom 9477 df-dju 9823 df-card 9861 df-gch 10542

This theorem is referenced by: (None)

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