Theorem r1sdom 9662

Description: Each stage in the cumulative hierarchy is strictly larger than the last. (Contributed by Mario Carneiro, 19-Apr-2013.)

Assertion

Ref	Expression
r1sdom	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → (𝑅1‘𝐵) ≺ (𝑅1‘𝐴))

Proof of Theorem r1sdom

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2820	. . . 4 ⊢ (𝑥 = ∅ → (𝐵 ∈ 𝑥 ↔ 𝐵 ∈ ∅))
2		fveq2 6817	. . . . 5 ⊢ (𝑥 = ∅ → (𝑅1‘𝑥) = (𝑅1‘∅))
3	2	breq2d 5098	. . . 4 ⊢ (𝑥 = ∅ → ((𝑅1‘𝐵) ≺ (𝑅1‘𝑥) ↔ (𝑅1‘𝐵) ≺ (𝑅1‘∅)))
4	1, 3	imbi12d 344	. . 3 ⊢ (𝑥 = ∅ → ((𝐵 ∈ 𝑥 → (𝑅1‘𝐵) ≺ (𝑅1‘𝑥)) ↔ (𝐵 ∈ ∅ → (𝑅1‘𝐵) ≺ (𝑅1‘∅))))
5		eleq2 2820	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐵 ∈ 𝑥 ↔ 𝐵 ∈ 𝑦))
6		fveq2 6817	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑅1‘𝑥) = (𝑅1‘𝑦))
7	6	breq2d 5098	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑅1‘𝐵) ≺ (𝑅1‘𝑥) ↔ (𝑅1‘𝐵) ≺ (𝑅1‘𝑦)))
8	5, 7	imbi12d 344	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐵 ∈ 𝑥 → (𝑅1‘𝐵) ≺ (𝑅1‘𝑥)) ↔ (𝐵 ∈ 𝑦 → (𝑅1‘𝐵) ≺ (𝑅1‘𝑦))))
9		eleq2 2820	. . . 4 ⊢ (𝑥 = suc 𝑦 → (𝐵 ∈ 𝑥 ↔ 𝐵 ∈ suc 𝑦))
10		fveq2 6817	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝑅1‘𝑥) = (𝑅1‘suc 𝑦))
11	10	breq2d 5098	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝑅1‘𝐵) ≺ (𝑅1‘𝑥) ↔ (𝑅1‘𝐵) ≺ (𝑅1‘suc 𝑦)))
12	9, 11	imbi12d 344	. . 3 ⊢ (𝑥 = suc 𝑦 → ((𝐵 ∈ 𝑥 → (𝑅1‘𝐵) ≺ (𝑅1‘𝑥)) ↔ (𝐵 ∈ suc 𝑦 → (𝑅1‘𝐵) ≺ (𝑅1‘suc 𝑦))))
13		eleq2 2820	. . . 4 ⊢ (𝑥 = 𝐴 → (𝐵 ∈ 𝑥 ↔ 𝐵 ∈ 𝐴))
14		fveq2 6817	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑅1‘𝑥) = (𝑅1‘𝐴))
15	14	breq2d 5098	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑅1‘𝐵) ≺ (𝑅1‘𝑥) ↔ (𝑅1‘𝐵) ≺ (𝑅1‘𝐴)))
16	13, 15	imbi12d 344	. . 3 ⊢ (𝑥 = 𝐴 → ((𝐵 ∈ 𝑥 → (𝑅1‘𝐵) ≺ (𝑅1‘𝑥)) ↔ (𝐵 ∈ 𝐴 → (𝑅1‘𝐵) ≺ (𝑅1‘𝐴))))
17		noel 4283	. . . 4 ⊢ ¬ 𝐵 ∈ ∅
18	17	pm2.21i 119	. . 3 ⊢ (𝐵 ∈ ∅ → (𝑅1‘𝐵) ≺ (𝑅1‘∅))
19		elsuci 6370	. . . . 5 ⊢ (𝐵 ∈ suc 𝑦 → (𝐵 ∈ 𝑦 ∨ 𝐵 = 𝑦))
20		sdomtr 9023	. . . . . . . . 9 ⊢ (((𝑅1‘𝐵) ≺ (𝑅1‘𝑦) ∧ (𝑅1‘𝑦) ≺ (𝑅1‘suc 𝑦)) → (𝑅1‘𝐵) ≺ (𝑅1‘suc 𝑦))
21	20	expcom 413	. . . . . . . 8 ⊢ ((𝑅1‘𝑦) ≺ (𝑅1‘suc 𝑦) → ((𝑅1‘𝐵) ≺ (𝑅1‘𝑦) → (𝑅1‘𝐵) ≺ (𝑅1‘suc 𝑦)))
22		fvex 6830	. . . . . . . . . 10 ⊢ (𝑅1‘𝑦) ∈ V
23	22	canth2 9038	. . . . . . . . 9 ⊢ (𝑅1‘𝑦) ≺ 𝒫 (𝑅1‘𝑦)
24		r1suc 9658	. . . . . . . . 9 ⊢ (𝑦 ∈ On → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1‘𝑦))
25	23, 24	breqtrrid 5124	. . . . . . . 8 ⊢ (𝑦 ∈ On → (𝑅1‘𝑦) ≺ (𝑅1‘suc 𝑦))
26	21, 25	syl11 33	. . . . . . 7 ⊢ ((𝑅1‘𝐵) ≺ (𝑅1‘𝑦) → (𝑦 ∈ On → (𝑅1‘𝐵) ≺ (𝑅1‘suc 𝑦)))
27	26	imim2i 16	. . . . . 6 ⊢ ((𝐵 ∈ 𝑦 → (𝑅1‘𝐵) ≺ (𝑅1‘𝑦)) → (𝐵 ∈ 𝑦 → (𝑦 ∈ On → (𝑅1‘𝐵) ≺ (𝑅1‘suc 𝑦))))
28		fveq2 6817	. . . . . . . . 9 ⊢ (𝐵 = 𝑦 → (𝑅1‘𝐵) = (𝑅1‘𝑦))
29	28	breq1d 5096	. . . . . . . 8 ⊢ (𝐵 = 𝑦 → ((𝑅1‘𝐵) ≺ (𝑅1‘suc 𝑦) ↔ (𝑅1‘𝑦) ≺ (𝑅1‘suc 𝑦)))
30	25, 29	imbitrrid 246	. . . . . . 7 ⊢ (𝐵 = 𝑦 → (𝑦 ∈ On → (𝑅1‘𝐵) ≺ (𝑅1‘suc 𝑦)))
31	30	a1i 11	. . . . . 6 ⊢ ((𝐵 ∈ 𝑦 → (𝑅1‘𝐵) ≺ (𝑅1‘𝑦)) → (𝐵 = 𝑦 → (𝑦 ∈ On → (𝑅1‘𝐵) ≺ (𝑅1‘suc 𝑦))))
32	27, 31	jaod 859	. . . . 5 ⊢ ((𝐵 ∈ 𝑦 → (𝑅1‘𝐵) ≺ (𝑅1‘𝑦)) → ((𝐵 ∈ 𝑦 ∨ 𝐵 = 𝑦) → (𝑦 ∈ On → (𝑅1‘𝐵) ≺ (𝑅1‘suc 𝑦))))
33	19, 32	syl5 34	. . . 4 ⊢ ((𝐵 ∈ 𝑦 → (𝑅1‘𝐵) ≺ (𝑅1‘𝑦)) → (𝐵 ∈ suc 𝑦 → (𝑦 ∈ On → (𝑅1‘𝐵) ≺ (𝑅1‘suc 𝑦))))
34	33	com3r 87	. . 3 ⊢ (𝑦 ∈ On → ((𝐵 ∈ 𝑦 → (𝑅1‘𝐵) ≺ (𝑅1‘𝑦)) → (𝐵 ∈ suc 𝑦 → (𝑅1‘𝐵) ≺ (𝑅1‘suc 𝑦))))
35		limuni 6363	. . . . . . 7 ⊢ (Lim 𝑥 → 𝑥 = ∪ 𝑥)
36	35	eleq2d 2817	. . . . . 6 ⊢ (Lim 𝑥 → (𝐵 ∈ 𝑥 ↔ 𝐵 ∈ ∪ 𝑥))
37		eluni2 4858	. . . . . 6 ⊢ (𝐵 ∈ ∪ 𝑥 ↔ ∃𝑦 ∈ 𝑥 𝐵 ∈ 𝑦)
38	36, 37	bitrdi 287	. . . . 5 ⊢ (Lim 𝑥 → (𝐵 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝑥 𝐵 ∈ 𝑦))
39		r19.29 3095	. . . . . . 7 ⊢ ((∀𝑦 ∈ 𝑥 (𝐵 ∈ 𝑦 → (𝑅1‘𝐵) ≺ (𝑅1‘𝑦)) ∧ ∃𝑦 ∈ 𝑥 𝐵 ∈ 𝑦) → ∃𝑦 ∈ 𝑥 ((𝐵 ∈ 𝑦 → (𝑅1‘𝐵) ≺ (𝑅1‘𝑦)) ∧ 𝐵 ∈ 𝑦))
40		fvex 6830	. . . . . . . . . 10 ⊢ (𝑅1‘𝑥) ∈ V
41		ssiun2 4991	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑥 → (𝑅1‘𝑦) ⊆ ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦))
42		vex 3440	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
43		r1lim 9660	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (𝑅1‘𝑥) = ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦))
44	42, 43	mpan 690	. . . . . . . . . . . 12 ⊢ (Lim 𝑥 → (𝑅1‘𝑥) = ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦))
45	44	sseq2d 3962	. . . . . . . . . . 11 ⊢ (Lim 𝑥 → ((𝑅1‘𝑦) ⊆ (𝑅1‘𝑥) ↔ (𝑅1‘𝑦) ⊆ ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦)))
46	41, 45	imbitrrid 246	. . . . . . . . . 10 ⊢ (Lim 𝑥 → (𝑦 ∈ 𝑥 → (𝑅1‘𝑦) ⊆ (𝑅1‘𝑥)))
47		ssdomg 8917	. . . . . . . . . 10 ⊢ ((𝑅1‘𝑥) ∈ V → ((𝑅1‘𝑦) ⊆ (𝑅1‘𝑥) → (𝑅1‘𝑦) ≼ (𝑅1‘𝑥)))
48	40, 46, 47	mpsylsyld 69	. . . . . . . . 9 ⊢ (Lim 𝑥 → (𝑦 ∈ 𝑥 → (𝑅1‘𝑦) ≼ (𝑅1‘𝑥)))
49		id 22	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑦 → (𝑅1‘𝐵) ≺ (𝑅1‘𝑦)) → (𝐵 ∈ 𝑦 → (𝑅1‘𝐵) ≺ (𝑅1‘𝑦)))
50	49	imp 406	. . . . . . . . . 10 ⊢ (((𝐵 ∈ 𝑦 → (𝑅1‘𝐵) ≺ (𝑅1‘𝑦)) ∧ 𝐵 ∈ 𝑦) → (𝑅1‘𝐵) ≺ (𝑅1‘𝑦))
51		sdomdomtr 9018	. . . . . . . . . . 11 ⊢ (((𝑅1‘𝐵) ≺ (𝑅1‘𝑦) ∧ (𝑅1‘𝑦) ≼ (𝑅1‘𝑥)) → (𝑅1‘𝐵) ≺ (𝑅1‘𝑥))
52	51	expcom 413	. . . . . . . . . 10 ⊢ ((𝑅1‘𝑦) ≼ (𝑅1‘𝑥) → ((𝑅1‘𝐵) ≺ (𝑅1‘𝑦) → (𝑅1‘𝐵) ≺ (𝑅1‘𝑥)))
53	50, 52	syl5 34	. . . . . . . . 9 ⊢ ((𝑅1‘𝑦) ≼ (𝑅1‘𝑥) → (((𝐵 ∈ 𝑦 → (𝑅1‘𝐵) ≺ (𝑅1‘𝑦)) ∧ 𝐵 ∈ 𝑦) → (𝑅1‘𝐵) ≺ (𝑅1‘𝑥)))
54	48, 53	syl6 35	. . . . . . . 8 ⊢ (Lim 𝑥 → (𝑦 ∈ 𝑥 → (((𝐵 ∈ 𝑦 → (𝑅1‘𝐵) ≺ (𝑅1‘𝑦)) ∧ 𝐵 ∈ 𝑦) → (𝑅1‘𝐵) ≺ (𝑅1‘𝑥))))
55	54	rexlimdv 3131	. . . . . . 7 ⊢ (Lim 𝑥 → (∃𝑦 ∈ 𝑥 ((𝐵 ∈ 𝑦 → (𝑅1‘𝐵) ≺ (𝑅1‘𝑦)) ∧ 𝐵 ∈ 𝑦) → (𝑅1‘𝐵) ≺ (𝑅1‘𝑥)))
56	39, 55	syl5 34	. . . . . 6 ⊢ (Lim 𝑥 → ((∀𝑦 ∈ 𝑥 (𝐵 ∈ 𝑦 → (𝑅1‘𝐵) ≺ (𝑅1‘𝑦)) ∧ ∃𝑦 ∈ 𝑥 𝐵 ∈ 𝑦) → (𝑅1‘𝐵) ≺ (𝑅1‘𝑥)))
57	56	expcomd 416	. . . . 5 ⊢ (Lim 𝑥 → (∃𝑦 ∈ 𝑥 𝐵 ∈ 𝑦 → (∀𝑦 ∈ 𝑥 (𝐵 ∈ 𝑦 → (𝑅1‘𝐵) ≺ (𝑅1‘𝑦)) → (𝑅1‘𝐵) ≺ (𝑅1‘𝑥))))
58	38, 57	sylbid 240	. . . 4 ⊢ (Lim 𝑥 → (𝐵 ∈ 𝑥 → (∀𝑦 ∈ 𝑥 (𝐵 ∈ 𝑦 → (𝑅1‘𝐵) ≺ (𝑅1‘𝑦)) → (𝑅1‘𝐵) ≺ (𝑅1‘𝑥))))
59	58	com23 86	. . 3 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (𝐵 ∈ 𝑦 → (𝑅1‘𝐵) ≺ (𝑅1‘𝑦)) → (𝐵 ∈ 𝑥 → (𝑅1‘𝐵) ≺ (𝑅1‘𝑥))))
60	4, 8, 12, 16, 18, 34, 59	tfinds 7785	. 2 ⊢ (𝐴 ∈ On → (𝐵 ∈ 𝐴 → (𝑅1‘𝐵) ≺ (𝑅1‘𝐴)))
61	60	imp 406	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → (𝑅1‘𝐵) ≺ (𝑅1‘𝐴))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056  Vcvv 3436   ⊆ wss 3897  ∅c0 4278  𝒫 cpw 4545  ∪ cuni 4854  ∪ ciun 4936   class class class wbr 5086  Oncon0 6301  Lim wlim 6302  suc csuc 6303  ‘cfv 6476   ≼ cdom 8862   ≺ csdm 8863  𝑅₁cr1 9650

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-ov 7344  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-er 8617  df-en 8865  df-dom 8866  df-sdom 8867  df-r1 9652

This theorem is referenced by:  r111  9663  smobeth  10472  r1tskina  10668