Theorem r1sdom 9816

Description: Each stage of the cumulative hierarchy of sets is strictly larger than any previous stage. (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 2853	. . . 4 ⊢ (𝑥 = ∅ → (𝐵 ∈ 𝑥 ↔ 𝐵 ∈ ∅))
2		fveq2 6914	. . . . 5 ⊢ (𝑥 = ∅ → (𝑅1‘𝑥) = (𝑅1‘∅))
3	2	breq2d 5118	. . . 4 ⊢ (𝑥 = ∅ → ((𝑅1‘𝐵) ≺ (𝑅1‘𝑥) ↔ (𝑅1‘𝐵) ≺ (𝑅1‘∅)))
4	1, 3	imbi12d 347	. . 3 ⊢ (𝑥 = ∅ → ((𝐵 ∈ 𝑥 → (𝑅1‘𝐵) ≺ (𝑅1‘𝑥)) ↔ (𝐵 ∈ ∅ → (𝑅1‘𝐵) ≺ (𝑅1‘∅))))
5		eleq2 2853	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐵 ∈ 𝑥 ↔ 𝐵 ∈ 𝑦))
6		fveq2 6914	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑅1‘𝑥) = (𝑅1‘𝑦))
7	6	breq2d 5118	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑅1‘𝐵) ≺ (𝑅1‘𝑥) ↔ (𝑅1‘𝐵) ≺ (𝑅1‘𝑦)))
8	5, 7	imbi12d 347	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐵 ∈ 𝑥 → (𝑅1‘𝐵) ≺ (𝑅1‘𝑥)) ↔ (𝐵 ∈ 𝑦 → (𝑅1‘𝐵) ≺ (𝑅1‘𝑦))))
9		eleq2 2853	. . . 4 ⊢ (𝑥 = suc 𝑦 → (𝐵 ∈ 𝑥 ↔ 𝐵 ∈ suc 𝑦))
10		fveq2 6914	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝑅1‘𝑥) = (𝑅1‘suc 𝑦))
11	10	breq2d 5118	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝑅1‘𝐵) ≺ (𝑅1‘𝑥) ↔ (𝑅1‘𝐵) ≺ (𝑅1‘suc 𝑦)))
12	9, 11	imbi12d 347	. . 3 ⊢ (𝑥 = suc 𝑦 → ((𝐵 ∈ 𝑥 → (𝑅1‘𝐵) ≺ (𝑅1‘𝑥)) ↔ (𝐵 ∈ suc 𝑦 → (𝑅1‘𝐵) ≺ (𝑅1‘suc 𝑦))))
13		eleq2 2853	. . . 4 ⊢ (𝑥 = 𝐴 → (𝐵 ∈ 𝑥 ↔ 𝐵 ∈ 𝐴))
14		fveq2 6914	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑅1‘𝑥) = (𝑅1‘𝐴))
15	14	breq2d 5118	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑅1‘𝐵) ≺ (𝑅1‘𝑥) ↔ (𝑅1‘𝐵) ≺ (𝑅1‘𝐴)))
16	13, 15	imbi12d 347	. . 3 ⊢ (𝑥 = 𝐴 → ((𝐵 ∈ 𝑥 → (𝑅1‘𝐵) ≺ (𝑅1‘𝑥)) ↔ (𝐵 ∈ 𝐴 → (𝑅1‘𝐵) ≺ (𝑅1‘𝐴))))
17		noel 4286	. . . 4 ⊢ ¬ 𝐵 ∈ ∅
18	17	pm2.21i 120	. . 3 ⊢ (𝐵 ∈ ∅ → (𝑅1‘𝐵) ≺ (𝑅1‘∅))
19		elsuci 6456	. . . . 5 ⊢ (𝐵 ∈ suc 𝑦 → (𝐵 ∈ 𝑦 ∨ 𝐵 = 𝑦))
20		sdomtr 9170	. . . . . . . . 9 ⊢ (((𝑅1‘𝐵) ≺ (𝑅1‘𝑦) ∧ (𝑅1‘𝑦) ≺ (𝑅1‘suc 𝑦)) → (𝑅1‘𝐵) ≺ (𝑅1‘suc 𝑦))
21	20	expcom 419	. . . . . . . 8 ⊢ ((𝑅1‘𝑦) ≺ (𝑅1‘suc 𝑦) → ((𝑅1‘𝐵) ≺ (𝑅1‘𝑦) → (𝑅1‘𝐵) ≺ (𝑅1‘suc 𝑦)))
22		fvex 6927	. . . . . . . . . 10 ⊢ (𝑅1‘𝑦) ∈ V
23	22	canth2 9185	. . . . . . . . 9 ⊢ (𝑅1‘𝑦) ≺ 𝒫 (𝑅1‘𝑦)
24		r1suc 9812	. . . . . . . . 9 ⊢ (𝑦 ∈ On → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1‘𝑦))
25	23, 24	breqtrrid 5147	. . . . . . . 8 ⊢ (𝑦 ∈ On → (𝑅1‘𝑦) ≺ (𝑅1‘suc 𝑦))
26	21, 25	syl11 34	. . . . . . 7 ⊢ ((𝑅1‘𝐵) ≺ (𝑅1‘𝑦) → (𝑦 ∈ On → (𝑅1‘𝐵) ≺ (𝑅1‘suc 𝑦)))
27	26	imim2i 17	. . . . . 6 ⊢ ((𝐵 ∈ 𝑦 → (𝑅1‘𝐵) ≺ (𝑅1‘𝑦)) → (𝐵 ∈ 𝑦 → (𝑦 ∈ On → (𝑅1‘𝐵) ≺ (𝑅1‘suc 𝑦))))
28		fveq2 6914	. . . . . . . . 9 ⊢ (𝐵 = 𝑦 → (𝑅1‘𝐵) = (𝑅1‘𝑦))
29	28	breq1d 5116	. . . . . . . 8 ⊢ (𝐵 = 𝑦 → ((𝑅1‘𝐵) ≺ (𝑅1‘suc 𝑦) ↔ (𝑅1‘𝑦) ≺ (𝑅1‘suc 𝑦)))
30	25, 29	imbitrrid 249	. . . . . . 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 873	. . . . 5 ⊢ ((𝐵 ∈ 𝑦 → (𝑅1‘𝐵) ≺ (𝑅1‘𝑦)) → ((𝐵 ∈ 𝑦 ∨ 𝐵 = 𝑦) → (𝑦 ∈ On → (𝑅1‘𝐵) ≺ (𝑅1‘suc 𝑦))))
33	19, 32	syl5 35	. . . 4 ⊢ ((𝐵 ∈ 𝑦 → (𝑅1‘𝐵) ≺ (𝑅1‘𝑦)) → (𝐵 ∈ suc 𝑦 → (𝑦 ∈ On → (𝑅1‘𝐵) ≺ (𝑅1‘suc 𝑦))))
34	33	com3r 88	. . 3 ⊢ (𝑦 ∈ On → ((𝐵 ∈ 𝑦 → (𝑅1‘𝐵) ≺ (𝑅1‘𝑦)) → (𝐵 ∈ suc 𝑦 → (𝑅1‘𝐵) ≺ (𝑅1‘suc 𝑦))))
35		limuni 6449	. . . . . . 7 ⊢ (Lim 𝑥 → 𝑥 = ∪ 𝑥)
36	35	eleq2d 2850	. . . . . 6 ⊢ (Lim 𝑥 → (𝐵 ∈ 𝑥 ↔ 𝐵 ∈ ∪ 𝑥))
37		eluni2 4874	. . . . . 6 ⊢ (𝐵 ∈ ∪ 𝑥 ↔ ∃𝑦 ∈ 𝑥 𝐵 ∈ 𝑦)
38	36, 37	bitrdi 290	. . . . 5 ⊢ (Lim 𝑥 → (𝐵 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝑥 𝐵 ∈ 𝑦))
39		r19.29 3129	. . . . . . 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 6927	. . . . . . . . . 10 ⊢ (𝑅1‘𝑥) ∈ V
41		ssiun2 5009	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑥 → (𝑅1‘𝑦) ⊆ ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦))
42		vex 3458	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
43		r1lim 9814	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (𝑅1‘𝑥) = ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦))
44	42, 43	mpan 703	. . . . . . . . . . . 12 ⊢ (Lim 𝑥 → (𝑅1‘𝑥) = ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦))
45	44	sseq2d 3945	. . . . . . . . . . 11 ⊢ (Lim 𝑥 → ((𝑅1‘𝑦) ⊆ (𝑅1‘𝑥) ↔ (𝑅1‘𝑦) ⊆ ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦)))
46	41, 45	imbitrrid 249	. . . . . . . . . 10 ⊢ (Lim 𝑥 → (𝑦 ∈ 𝑥 → (𝑅1‘𝑦) ⊆ (𝑅1‘𝑥)))
47		ssdomg 9063	. . . . . . . . . 10 ⊢ ((𝑅1‘𝑥) ∈ V → ((𝑅1‘𝑦) ⊆ (𝑅1‘𝑥) → (𝑅1‘𝑦) ≼ (𝑅1‘𝑥)))
48	40, 46, 47	mpsylsyld 70	. . . . . . . . 9 ⊢ (Lim 𝑥 → (𝑦 ∈ 𝑥 → (𝑅1‘𝑦) ≼ (𝑅1‘𝑥)))
49		id 23	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑦 → (𝑅1‘𝐵) ≺ (𝑅1‘𝑦)) → (𝐵 ∈ 𝑦 → (𝑅1‘𝐵) ≺ (𝑅1‘𝑦)))
50	49	imp 412	. . . . . . . . . 10 ⊢ (((𝐵 ∈ 𝑦 → (𝑅1‘𝐵) ≺ (𝑅1‘𝑦)) ∧ 𝐵 ∈ 𝑦) → (𝑅1‘𝐵) ≺ (𝑅1‘𝑦))
51		sdomdomtr 9165	. . . . . . . . . . 11 ⊢ (((𝑅1‘𝐵) ≺ (𝑅1‘𝑦) ∧ (𝑅1‘𝑦) ≼ (𝑅1‘𝑥)) → (𝑅1‘𝐵) ≺ (𝑅1‘𝑥))
52	51	expcom 419	. . . . . . . . . 10 ⊢ ((𝑅1‘𝑦) ≼ (𝑅1‘𝑥) → ((𝑅1‘𝐵) ≺ (𝑅1‘𝑦) → (𝑅1‘𝐵) ≺ (𝑅1‘𝑥)))
53	50, 52	syl5 35	. . . . . . . . 9 ⊢ ((𝑅1‘𝑦) ≼ (𝑅1‘𝑥) → (((𝐵 ∈ 𝑦 → (𝑅1‘𝐵) ≺ (𝑅1‘𝑦)) ∧ 𝐵 ∈ 𝑦) → (𝑅1‘𝐵) ≺ (𝑅1‘𝑥)))
54	48, 53	syl6 36	. . . . . . . 8 ⊢ (Lim 𝑥 → (𝑦 ∈ 𝑥 → (((𝐵 ∈ 𝑦 → (𝑅1‘𝐵) ≺ (𝑅1‘𝑦)) ∧ 𝐵 ∈ 𝑦) → (𝑅1‘𝐵) ≺ (𝑅1‘𝑥))))
55	54	rexlimdv 3165	. . . . . . 7 ⊢ (Lim 𝑥 → (∃𝑦 ∈ 𝑥 ((𝐵 ∈ 𝑦 → (𝑅1‘𝐵) ≺ (𝑅1‘𝑦)) ∧ 𝐵 ∈ 𝑦) → (𝑅1‘𝐵) ≺ (𝑅1‘𝑥)))
56	39, 55	syl5 35	. . . . . 6 ⊢ (Lim 𝑥 → ((∀𝑦 ∈ 𝑥 (𝐵 ∈ 𝑦 → (𝑅1‘𝐵) ≺ (𝑅1‘𝑦)) ∧ ∃𝑦 ∈ 𝑥 𝐵 ∈ 𝑦) → (𝑅1‘𝐵) ≺ (𝑅1‘𝑥)))
57	56	expcomd 422	. . . . 5 ⊢ (Lim 𝑥 → (∃𝑦 ∈ 𝑥 𝐵 ∈ 𝑦 → (∀𝑦 ∈ 𝑥 (𝐵 ∈ 𝑦 → (𝑅1‘𝐵) ≺ (𝑅1‘𝑦)) → (𝑅1‘𝐵) ≺ (𝑅1‘𝑥))))
58	38, 57	sylbid 243	. . . 4 ⊢ (Lim 𝑥 → (𝐵 ∈ 𝑥 → (∀𝑦 ∈ 𝑥 (𝐵 ∈ 𝑦 → (𝑅1‘𝐵) ≺ (𝑅1‘𝑦)) → (𝑅1‘𝐵) ≺ (𝑅1‘𝑥))))
59	58	com23 87	. . 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 7903	. 2 ⊢ (𝐴 ∈ On → (𝐵 ∈ 𝐴 → (𝑅1‘𝐵) ≺ (𝑅1‘𝐴)))
61	60	imp 412	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → (𝑅1‘𝐵) ≺ (𝑅1‘𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900  ∅c0 4281  𝒫 cpw 4560  ∪ cuni 4870  ∪ ciun 4954   class class class wbr 5106  Oncon0 6386  Lim wlim 6387  suc csuc 6388  ‘cfv 6564   ≼ cdom 9007   ≺ csdm 9008  𝑅₁cr1 9801

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7781

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6328  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6518  df-fun 6566  df-fn 6567  df-f 6568  df-f1 6569  df-fo 6570  df-f1o 6571  df-fv 6572  df-ov 7452  df-2nd 8034  df-frecs 8335  df-wrecs 8366  df-recs 8415  df-rdg 8454  df-er 8753  df-en 9010  df-dom 9011  df-sdom 9012  df-r1 9803

This theorem is used by:  r111  9817  smobeth  10708  r1tskina  10904