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Theorem gchaclem 10576

Description: Lemma for gchac 10579 (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 494	. . 3 ⊢ (𝜑 → 𝐴 ≼ 𝐶)
3		reldom 8881	. . . . . 6 ⊢ Rel ≼
4	3	brrelex2i 5676	. . . . 5 ⊢ (𝐴 ≼ 𝐶 → 𝐶 ∈ V)
5	2, 4	syl 17	. . . 4 ⊢ (𝜑 → 𝐶 ∈ V)
6		canth2g 9051	. . . 4 ⊢ (𝐶 ∈ V → 𝐶 ≺ 𝒫 𝐶)
7		sdomdom 8909	. . . 4 ⊢ (𝐶 ≺ 𝒫 𝐶 → 𝐶 ≼ 𝒫 𝐶)
8	5, 6, 7	3syl 18	. . 3 ⊢ (𝜑 → 𝐶 ≼ 𝒫 𝐶)
9		domtr 8936	. . 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 8936	. . . . . . . 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 10090	. . . . . 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 10527	. . . . 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 1373	. . . 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 9049	. . . . 5 ⊢ (𝐴 ≼ 𝐶 → 𝒫 𝐴 ≼ 𝒫 𝐶)
24		domtr 8936	. . . . . 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 2113 Vcvv 3437 𝒫 cpw 4549 class class class wbr 5093 ωcom 7802 ≈ cen 8872 ≼ cdom 8873 ≺ csdm 8874 ⊔ cdju 9798 GCHcgch 10518

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-ral 3049 df-rex 3058 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-int 4898 df-iun 4943 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7803 df-1st 7927 df-2nd 7928 df-1o 8391 df-2o 8392 df-er 8628 df-map 8758 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-wdom 9458 df-dju 9801 df-card 9839 df-gch 10519

This theorem is referenced by: (None)

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