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Theorem gchaclem 9700

Description: Lemma for gchac 9703 (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 478	. . 3 ⊢ (𝜑 → 𝐴 ≼ 𝐶)
3		reldom 8113	. . . . . 6 ⊢ Rel ≼
4	3	brrelex2i 5298	. . . . 5 ⊢ (𝐴 ≼ 𝐶 → 𝐶 ∈ V)
5	2, 4	syl 17	. . . 4 ⊢ (𝜑 → 𝐶 ∈ V)
6		canth2g 8268	. . . 4 ⊢ (𝐶 ∈ V → 𝐶 ≺ 𝒫 𝐶)
7		sdomdom 8135	. . . 4 ⊢ (𝐶 ≺ 𝒫 𝐶 → 𝐶 ≼ 𝒫 𝐶)
8	5, 6, 7	3syl 18	. . 3 ⊢ (𝜑 → 𝐶 ≼ 𝒫 𝐶)
9		domtr 8160	. . 3 ⊢ ((𝐴 ≼ 𝐶 ∧ 𝐶 ≼ 𝒫 𝐶) → 𝐴 ≼ 𝒫 𝐶)
10	2, 8, 9	syl2anc 693	. 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 8160	. . . . . . . 8 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ≼ 𝐶) → ω ≼ 𝐶)
15	13, 2, 14	syl2anc 693	. . . . . . 7 ⊢ (𝜑 → ω ≼ 𝐶)
16	15	adantr 473	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≼ 𝒫 𝒫 𝐶) → ω ≼ 𝐶)
17		pwcdaidm 9217	. . . . . 6 ⊢ (ω ≼ 𝐶 → (𝒫 𝐶 +_𝑐 𝒫 𝐶) ≈ 𝒫 𝐶)
18	16, 17	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≼ 𝒫 𝒫 𝐶) → (𝒫 𝐶 +_𝑐 𝒫 𝐶) ≈ 𝒫 𝐶)
19		simpr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≼ 𝒫 𝒫 𝐶) → 𝐵 ≼ 𝒫 𝒫 𝐶)
20		gchdomtri 9651	. . . . 5 ⊢ ((𝒫 𝐶 ∈ GCH ∧ (𝒫 𝐶 +_𝑐 𝒫 𝐶) ≈ 𝒫 𝐶 ∧ 𝐵 ≼ 𝒫 𝒫 𝐶) → (𝒫 𝐶 ≼ 𝐵 ∨ 𝐵 ≼ 𝒫 𝐶))
21	12, 18, 19, 20	syl3anc 1474	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≼ 𝒫 𝒫 𝐶) → (𝒫 𝐶 ≼ 𝐵 ∨ 𝐵 ≼ 𝒫 𝐶))
22	21	ex 448	. . 3 ⊢ (𝜑 → (𝐵 ≼ 𝒫 𝒫 𝐶 → (𝒫 𝐶 ≼ 𝐵 ∨ 𝐵 ≼ 𝒫 𝐶)))
23		pwdom 8266	. . . . 5 ⊢ (𝐴 ≼ 𝐶 → 𝒫 𝐴 ≼ 𝒫 𝐶)
24		domtr 8160	. . . . . 6 ⊢ ((𝒫 𝐴 ≼ 𝒫 𝐶 ∧ 𝒫 𝐶 ≼ 𝐵) → 𝒫 𝐴 ≼ 𝐵)
25	24	ex 448	. . . . 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 483	. . . 4 ⊢ (𝜑 → (𝐵 ≼ 𝒫 𝐶 → 𝒫 𝐴 ≼ 𝐵))
28	26, 27	jaod 394	. . 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 556	1 ⊢ (𝜑 → (𝐴 ≼ 𝒫 𝐶 ∧ (𝐵 ≼ 𝒫 𝒫 𝐶 → 𝒫 𝐴 ≼ 𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∨ wo 382 ∧ wa 383 ∈ wcel 2143 Vcvv 3348 𝒫 cpw 4294 class class class wbr 4783 (class class class)co 6791 ωcom 7210 ≈ cen 8104 ≼ cdom 8105 ≺ csdm 8106 +_𝑐 ccda 9189 GCHcgch 9642

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1868 ax-4 1883 ax-5 1989 ax-6 2055 ax-7 2091 ax-8 2145 ax-9 2152 ax-10 2172 ax-11 2188 ax-12 2201 ax-13 2406 ax-ext 2749 ax-sep 4911 ax-nul 4919 ax-pow 4970 ax-pr 5033 ax-un 7094

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1070 df-3an 1071 df-tru 1632 df-ex 1851 df-nf 1856 df-sb 2048 df-eu 2620 df-mo 2621 df-clab 2756 df-cleq 2762 df-clel 2765 df-nfc 2900 df-ne 2942 df-ral 3064 df-rex 3065 df-reu 3066 df-rab 3068 df-v 3350 df-sbc 3585 df-csb 3680 df-dif 3723 df-un 3725 df-in 3727 df-ss 3734 df-pss 3736 df-nul 4061 df-if 4223 df-pw 4296 df-sn 4314 df-pr 4316 df-tp 4318 df-op 4320 df-uni 4572 df-int 4609 df-iun 4653 df-br 4784 df-opab 4844 df-mpt 4861 df-tr 4884 df-id 5156 df-eprel 5161 df-po 5169 df-so 5170 df-fr 5207 df-we 5209 df-xp 5254 df-rel 5255 df-cnv 5256 df-co 5257 df-dm 5258 df-rn 5259 df-res 5260 df-ima 5261 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-ov 6794 df-oprab 6795 df-mpt2 6796 df-om 7211 df-1st 7313 df-2nd 7314 df-1o 7711 df-2o 7712 df-er 7894 df-map 8009 df-en 8108 df-dom 8109 df-sdom 8110 df-fin 8111 df-wdom 8618 df-card 8963 df-cda 9190 df-gch 9643

This theorem is referenced by: (None)

W3C validator