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Theorem gchaclem 9896

Description: Lemma for gchac 9899 (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 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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