Theorem canthp1lem1 9353

Description: Lemma for canthp1 9355. (Contributed by Mario Carneiro, 18-May-2015.)

Assertion

Ref	Expression
canthp1lem1	⊢ (1𝑜 ≺ 𝐴 → (𝐴 +𝑐 2𝑜) ≼ 𝒫 𝐴)

Proof of Theorem canthp1lem1

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1sdom2 8044	. . 3 ⊢ 1𝑜 ≺ 2𝑜
2		cdaxpdom 8894	. . 3 ⊢ ((1𝑜 ≺ 𝐴 ∧ 1𝑜 ≺ 2𝑜) → (𝐴 +𝑐 2𝑜) ≼ (𝐴 × 2𝑜))
3	1, 2	mpan2 703	. 2 ⊢ (1𝑜 ≺ 𝐴 → (𝐴 +𝑐 2𝑜) ≼ (𝐴 × 2𝑜))
4		sdom0 7977	. . . . . 6 ⊢ ¬ 1𝑜 ≺ ∅
5		breq2 4587	. . . . . 6 ⊢ (𝐴 = ∅ → (1𝑜 ≺ 𝐴 ↔ 1𝑜 ≺ ∅))
6	4, 5	mtbiri 316	. . . . 5 ⊢ (𝐴 = ∅ → ¬ 1𝑜 ≺ 𝐴)
7	6	con2i 133	. . . 4 ⊢ (1𝑜 ≺ 𝐴 → ¬ 𝐴 = ∅)
8		neq0 3889	. . . 4 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
9	7, 8	sylib 207	. . 3 ⊢ (1𝑜 ≺ 𝐴 → ∃𝑥 𝑥 ∈ 𝐴)
10		relsdom 7848	. . . . . . . . . 10 ⊢ Rel ≺
11	10	brrelex2i 5083	. . . . . . . . 9 ⊢ (1𝑜 ≺ 𝐴 → 𝐴 ∈ V)
12	11	adantr 480	. . . . . . . 8 ⊢ ((1𝑜 ≺ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝐴 ∈ V)
13		enrefg 7873	. . . . . . . 8 ⊢ (𝐴 ∈ V → 𝐴 ≈ 𝐴)
14	12, 13	syl 17	. . . . . . 7 ⊢ ((1𝑜 ≺ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝐴 ≈ 𝐴)
15		df2o2 7461	. . . . . . . . 9 ⊢ 2𝑜 = {∅, {∅}}
16		pwpw0 4284	. . . . . . . . 9 ⊢ 𝒫 {∅} = {∅, {∅}}
17	15, 16	eqtr4i 2635	. . . . . . . 8 ⊢ 2𝑜 = 𝒫 {∅}
18		0ex 4718	. . . . . . . . . 10 ⊢ ∅ ∈ V
19		vex 3176	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
20		en2sn 7922	. . . . . . . . . 10 ⊢ ((∅ ∈ V ∧ 𝑥 ∈ V) → {∅} ≈ {𝑥})
21	18, 19, 20	mp2an 704	. . . . . . . . 9 ⊢ {∅} ≈ {𝑥}
22		pwen 8018	. . . . . . . . 9 ⊢ ({∅} ≈ {𝑥} → 𝒫 {∅} ≈ 𝒫 {𝑥})
23	21, 22	ax-mp 5	. . . . . . . 8 ⊢ 𝒫 {∅} ≈ 𝒫 {𝑥}
24	17, 23	eqbrtri 4604	. . . . . . 7 ⊢ 2𝑜 ≈ 𝒫 {𝑥}
25		xpen 8008	. . . . . . 7 ⊢ ((𝐴 ≈ 𝐴 ∧ 2𝑜 ≈ 𝒫 {𝑥}) → (𝐴 × 2𝑜) ≈ (𝐴 × 𝒫 {𝑥}))
26	14, 24, 25	sylancl 693	. . . . . 6 ⊢ ((1𝑜 ≺ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐴 × 2𝑜) ≈ (𝐴 × 𝒫 {𝑥}))
27		snex 4835	. . . . . . . 8 ⊢ {𝑥} ∈ V
28	27	pwex 4774	. . . . . . 7 ⊢ 𝒫 {𝑥} ∈ V
29		uncom 3719	. . . . . . . . 9 ⊢ ((𝐴 ∖ {𝑥}) ∪ {𝑥}) = ({𝑥} ∪ (𝐴 ∖ {𝑥}))
30		simpr 476	. . . . . . . . . . 11 ⊢ ((1𝑜 ≺ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
31	30	snssd 4281	. . . . . . . . . 10 ⊢ ((1𝑜 ≺ 𝐴 ∧ 𝑥 ∈ 𝐴) → {𝑥} ⊆ 𝐴)
32		undif 4001	. . . . . . . . . 10 ⊢ ({𝑥} ⊆ 𝐴 ↔ ({𝑥} ∪ (𝐴 ∖ {𝑥})) = 𝐴)
33	31, 32	sylib 207	. . . . . . . . 9 ⊢ ((1𝑜 ≺ 𝐴 ∧ 𝑥 ∈ 𝐴) → ({𝑥} ∪ (𝐴 ∖ {𝑥})) = 𝐴)
34	29, 33	syl5eq 2656	. . . . . . . 8 ⊢ ((1𝑜 ≺ 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐴 ∖ {𝑥}) ∪ {𝑥}) = 𝐴)
35		difexg 4735	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (𝐴 ∖ {𝑥}) ∈ V)
36	12, 35	syl 17	. . . . . . . . 9 ⊢ ((1𝑜 ≺ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐴 ∖ {𝑥}) ∈ V)
37		canth2g 7999	. . . . . . . . 9 ⊢ ((𝐴 ∖ {𝑥}) ∈ V → (𝐴 ∖ {𝑥}) ≺ 𝒫 (𝐴 ∖ {𝑥}))
38		domunsn 7995	. . . . . . . . 9 ⊢ ((𝐴 ∖ {𝑥}) ≺ 𝒫 (𝐴 ∖ {𝑥}) → ((𝐴 ∖ {𝑥}) ∪ {𝑥}) ≼ 𝒫 (𝐴 ∖ {𝑥}))
39	36, 37, 38	3syl 18	. . . . . . . 8 ⊢ ((1𝑜 ≺ 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐴 ∖ {𝑥}) ∪ {𝑥}) ≼ 𝒫 (𝐴 ∖ {𝑥}))
40	34, 39	eqbrtrrd 4607	. . . . . . 7 ⊢ ((1𝑜 ≺ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝐴 ≼ 𝒫 (𝐴 ∖ {𝑥}))
41		xpdom1g 7942	. . . . . . 7 ⊢ ((𝒫 {𝑥} ∈ V ∧ 𝐴 ≼ 𝒫 (𝐴 ∖ {𝑥})) → (𝐴 × 𝒫 {𝑥}) ≼ (𝒫 (𝐴 ∖ {𝑥}) × 𝒫 {𝑥}))
42	28, 40, 41	sylancr 694	. . . . . 6 ⊢ ((1𝑜 ≺ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐴 × 𝒫 {𝑥}) ≼ (𝒫 (𝐴 ∖ {𝑥}) × 𝒫 {𝑥}))
43		endomtr 7900	. . . . . 6 ⊢ (((𝐴 × 2𝑜) ≈ (𝐴 × 𝒫 {𝑥}) ∧ (𝐴 × 𝒫 {𝑥}) ≼ (𝒫 (𝐴 ∖ {𝑥}) × 𝒫 {𝑥})) → (𝐴 × 2𝑜) ≼ (𝒫 (𝐴 ∖ {𝑥}) × 𝒫 {𝑥}))
44	26, 42, 43	syl2anc 691	. . . . 5 ⊢ ((1𝑜 ≺ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐴 × 2𝑜) ≼ (𝒫 (𝐴 ∖ {𝑥}) × 𝒫 {𝑥}))
45		pwcdaen 8890	. . . . . . 7 ⊢ (((𝐴 ∖ {𝑥}) ∈ V ∧ {𝑥} ∈ V) → 𝒫 ((𝐴 ∖ {𝑥}) +𝑐 {𝑥}) ≈ (𝒫 (𝐴 ∖ {𝑥}) × 𝒫 {𝑥}))
46	36, 27, 45	sylancl 693	. . . . . 6 ⊢ ((1𝑜 ≺ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝒫 ((𝐴 ∖ {𝑥}) +𝑐 {𝑥}) ≈ (𝒫 (𝐴 ∖ {𝑥}) × 𝒫 {𝑥}))
47	46	ensymd 7893	. . . . 5 ⊢ ((1𝑜 ≺ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝒫 (𝐴 ∖ {𝑥}) × 𝒫 {𝑥}) ≈ 𝒫 ((𝐴 ∖ {𝑥}) +𝑐 {𝑥}))
48		domentr 7901	. . . . 5 ⊢ (((𝐴 × 2𝑜) ≼ (𝒫 (𝐴 ∖ {𝑥}) × 𝒫 {𝑥}) ∧ (𝒫 (𝐴 ∖ {𝑥}) × 𝒫 {𝑥}) ≈ 𝒫 ((𝐴 ∖ {𝑥}) +𝑐 {𝑥})) → (𝐴 × 2𝑜) ≼ 𝒫 ((𝐴 ∖ {𝑥}) +𝑐 {𝑥}))
49	44, 47, 48	syl2anc 691	. . . 4 ⊢ ((1𝑜 ≺ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐴 × 2𝑜) ≼ 𝒫 ((𝐴 ∖ {𝑥}) +𝑐 {𝑥}))
50	27	a1i 11	. . . . . . 7 ⊢ ((1𝑜 ≺ 𝐴 ∧ 𝑥 ∈ 𝐴) → {𝑥} ∈ V)
51		incom 3767	. . . . . . . . 9 ⊢ ((𝐴 ∖ {𝑥}) ∩ {𝑥}) = ({𝑥} ∩ (𝐴 ∖ {𝑥}))
52		disjdif 3992	. . . . . . . . 9 ⊢ ({𝑥} ∩ (𝐴 ∖ {𝑥})) = ∅
53	51, 52	eqtri 2632	. . . . . . . 8 ⊢ ((𝐴 ∖ {𝑥}) ∩ {𝑥}) = ∅
54	53	a1i 11	. . . . . . 7 ⊢ ((1𝑜 ≺ 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐴 ∖ {𝑥}) ∩ {𝑥}) = ∅)
55		cdaun 8877	. . . . . . 7 ⊢ (((𝐴 ∖ {𝑥}) ∈ V ∧ {𝑥} ∈ V ∧ ((𝐴 ∖ {𝑥}) ∩ {𝑥}) = ∅) → ((𝐴 ∖ {𝑥}) +𝑐 {𝑥}) ≈ ((𝐴 ∖ {𝑥}) ∪ {𝑥}))
56	36, 50, 54, 55	syl3anc 1318	. . . . . 6 ⊢ ((1𝑜 ≺ 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐴 ∖ {𝑥}) +𝑐 {𝑥}) ≈ ((𝐴 ∖ {𝑥}) ∪ {𝑥}))
57	56, 34	breqtrd 4609	. . . . 5 ⊢ ((1𝑜 ≺ 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐴 ∖ {𝑥}) +𝑐 {𝑥}) ≈ 𝐴)
58		pwen 8018	. . . . 5 ⊢ (((𝐴 ∖ {𝑥}) +𝑐 {𝑥}) ≈ 𝐴 → 𝒫 ((𝐴 ∖ {𝑥}) +𝑐 {𝑥}) ≈ 𝒫 𝐴)
59	57, 58	syl 17	. . . 4 ⊢ ((1𝑜 ≺ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝒫 ((𝐴 ∖ {𝑥}) +𝑐 {𝑥}) ≈ 𝒫 𝐴)
60		domentr 7901	. . . 4 ⊢ (((𝐴 × 2𝑜) ≼ 𝒫 ((𝐴 ∖ {𝑥}) +𝑐 {𝑥}) ∧ 𝒫 ((𝐴 ∖ {𝑥}) +𝑐 {𝑥}) ≈ 𝒫 𝐴) → (𝐴 × 2𝑜) ≼ 𝒫 𝐴)
61	49, 59, 60	syl2anc 691	. . 3 ⊢ ((1𝑜 ≺ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐴 × 2𝑜) ≼ 𝒫 𝐴)
62	9, 61	exlimddv 1850	. 2 ⊢ (1𝑜 ≺ 𝐴 → (𝐴 × 2𝑜) ≼ 𝒫 𝐴)
63		domtr 7895	. 2 ⊢ (((𝐴 +𝑐 2𝑜) ≼ (𝐴 × 2𝑜) ∧ (𝐴 × 2𝑜) ≼ 𝒫 𝐴) → (𝐴 +𝑐 2𝑜) ≼ 𝒫 𝐴)
64	3, 62, 63	syl2anc 691	1 ⊢ (1𝑜 ≺ 𝐴 → (𝐴 +𝑐 2𝑜) ≼ 𝒫 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  {cpr 4127   class class class wbr 4583   × cxp 5036  (class class class)co 6549  1_𝑜c1o 7440  2_𝑜c2o 7441   ≈ cen 7838   ≼ cdom 7839   ≺ csdm 7840   +_𝑐 ccda 8872

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-1o 7447  df-2o 7448  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-cda 8873

This theorem is referenced by:  canthp1lem2  9354  canthp1  9355