Theorem r1sdom 9843

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 2833	. . . 4 ⊢ (𝑥 = ∅ → (𝐵 ∈ 𝑥 ↔ 𝐵 ∈ ∅))
2		fveq2 6920	. . . . 5 ⊢ (𝑥 = ∅ → (𝑅1‘𝑥) = (𝑅1‘∅))
3	2	breq2d 5178	. . . 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 2833	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐵 ∈ 𝑥 ↔ 𝐵 ∈ 𝑦))
6		fveq2 6920	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑅1‘𝑥) = (𝑅1‘𝑦))
7	6	breq2d 5178	. . . 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 2833	. . . 4 ⊢ (𝑥 = suc 𝑦 → (𝐵 ∈ 𝑥 ↔ 𝐵 ∈ suc 𝑦))
10		fveq2 6920	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝑅1‘𝑥) = (𝑅1‘suc 𝑦))
11	10	breq2d 5178	. . . 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 2833	. . . 4 ⊢ (𝑥 = 𝐴 → (𝐵 ∈ 𝑥 ↔ 𝐵 ∈ 𝐴))
14		fveq2 6920	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑅1‘𝑥) = (𝑅1‘𝐴))
15	14	breq2d 5178	. . . 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 4360	. . . 4 ⊢ ¬ 𝐵 ∈ ∅
18	17	pm2.21i 119	. . 3 ⊢ (𝐵 ∈ ∅ → (𝑅1‘𝐵) ≺ (𝑅1‘∅))
19		elsuci 6462	. . . . 5 ⊢ (𝐵 ∈ suc 𝑦 → (𝐵 ∈ 𝑦 ∨ 𝐵 = 𝑦))
20		sdomtr 9181	. . . . . . . . 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 6933	. . . . . . . . . 10 ⊢ (𝑅1‘𝑦) ∈ V
23	22	canth2 9196	. . . . . . . . 9 ⊢ (𝑅1‘𝑦) ≺ 𝒫 (𝑅1‘𝑦)
24		r1suc 9839	. . . . . . . . 9 ⊢ (𝑦 ∈ On → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1‘𝑦))
25	23, 24	breqtrrid 5204	. . . . . . . 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 6920	. . . . . . . . 9 ⊢ (𝐵 = 𝑦 → (𝑅1‘𝐵) = (𝑅1‘𝑦))
29	28	breq1d 5176	. . . . . . . 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 858	. . . . 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 6456	. . . . . . 7 ⊢ (Lim 𝑥 → 𝑥 = ∪ 𝑥)
36	35	eleq2d 2830	. . . . . 6 ⊢ (Lim 𝑥 → (𝐵 ∈ 𝑥 ↔ 𝐵 ∈ ∪ 𝑥))
37		eluni2 4935	. . . . . 6 ⊢ (𝐵 ∈ ∪ 𝑥 ↔ ∃𝑦 ∈ 𝑥 𝐵 ∈ 𝑦)
38	36, 37	bitrdi 287	. . . . 5 ⊢ (Lim 𝑥 → (𝐵 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝑥 𝐵 ∈ 𝑦))
39		r19.29 3120	. . . . . . 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 6933	. . . . . . . . . 10 ⊢ (𝑅1‘𝑥) ∈ V
41		ssiun2 5070	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑥 → (𝑅1‘𝑦) ⊆ ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦))
42		vex 3492	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
43		r1lim 9841	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (𝑅1‘𝑥) = ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦))
44	42, 43	mpan 689	. . . . . . . . . . . 12 ⊢ (Lim 𝑥 → (𝑅1‘𝑥) = ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦))
45	44	sseq2d 4041	. . . . . . . . . . 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 9060	. . . . . . . . . 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 9176	. . . . . . . . . . 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 3159	. . . . . . 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 7897	. 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 846   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076  Vcvv 3488   ⊆ wss 3976  ∅c0 4352  𝒫 cpw 4622  ∪ cuni 4931  ∪ ciun 5015   class class class wbr 5166  Oncon0 6395  Lim wlim 6396  suc csuc 6397  ‘cfv 6573   ≼ cdom 9001   ≺ csdm 9002  𝑅₁cr1 9831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-r1 9833

This theorem is referenced by:  r111  9844  smobeth  10655  r1tskina  10851