Theorem r1sdom 9734

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 2818	. . . 4 ⊢ (𝑥 = ∅ → (𝐵 ∈ 𝑥 ↔ 𝐵 ∈ ∅))
2		fveq2 6861	. . . . 5 ⊢ (𝑥 = ∅ → (𝑅1‘𝑥) = (𝑅1‘∅))
3	2	breq2d 5122	. . . 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 2818	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐵 ∈ 𝑥 ↔ 𝐵 ∈ 𝑦))
6		fveq2 6861	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑅1‘𝑥) = (𝑅1‘𝑦))
7	6	breq2d 5122	. . . 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 2818	. . . 4 ⊢ (𝑥 = suc 𝑦 → (𝐵 ∈ 𝑥 ↔ 𝐵 ∈ suc 𝑦))
10		fveq2 6861	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝑅1‘𝑥) = (𝑅1‘suc 𝑦))
11	10	breq2d 5122	. . . 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 2818	. . . 4 ⊢ (𝑥 = 𝐴 → (𝐵 ∈ 𝑥 ↔ 𝐵 ∈ 𝐴))
14		fveq2 6861	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑅1‘𝑥) = (𝑅1‘𝐴))
15	14	breq2d 5122	. . . 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 4304	. . . 4 ⊢ ¬ 𝐵 ∈ ∅
18	17	pm2.21i 119	. . 3 ⊢ (𝐵 ∈ ∅ → (𝑅1‘𝐵) ≺ (𝑅1‘∅))
19		elsuci 6404	. . . . 5 ⊢ (𝐵 ∈ suc 𝑦 → (𝐵 ∈ 𝑦 ∨ 𝐵 = 𝑦))
20		sdomtr 9085	. . . . . . . . 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 6874	. . . . . . . . . 10 ⊢ (𝑅1‘𝑦) ∈ V
23	22	canth2 9100	. . . . . . . . 9 ⊢ (𝑅1‘𝑦) ≺ 𝒫 (𝑅1‘𝑦)
24		r1suc 9730	. . . . . . . . 9 ⊢ (𝑦 ∈ On → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1‘𝑦))
25	23, 24	breqtrrid 5148	. . . . . . . 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 6861	. . . . . . . . 9 ⊢ (𝐵 = 𝑦 → (𝑅1‘𝐵) = (𝑅1‘𝑦))
29	28	breq1d 5120	. . . . . . . 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 6397	. . . . . . 7 ⊢ (Lim 𝑥 → 𝑥 = ∪ 𝑥)
36	35	eleq2d 2815	. . . . . 6 ⊢ (Lim 𝑥 → (𝐵 ∈ 𝑥 ↔ 𝐵 ∈ ∪ 𝑥))
37		eluni2 4878	. . . . . 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 6874	. . . . . . . . . 10 ⊢ (𝑅1‘𝑥) ∈ V
41		ssiun2 5014	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑥 → (𝑅1‘𝑦) ⊆ ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦))
42		vex 3454	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
43		r1lim 9732	. . . . . . . . . . . . 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 3982	. . . . . . . . . . 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 8974	. . . . . . . . . 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 9080	. . . . . . . . . . 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 3133	. . . . . . 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 7839	. 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 1540   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054  Vcvv 3450   ⊆ wss 3917  ∅c0 4299  𝒫 cpw 4566  ∪ cuni 4874  ∪ ciun 4958   class class class wbr 5110  Oncon0 6335  Lim wlim 6336  suc csuc 6337  ‘cfv 6514   ≼ cdom 8919   ≺ csdm 8920  𝑅₁cr1 9722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-er 8674  df-en 8922  df-dom 8923  df-sdom 8924  df-r1 9724

This theorem is referenced by:  r111  9735  smobeth  10546  r1tskina  10742