Theorem r111 9757

Description: The cumulative hierarchy is a one-to-one function. (Contributed by Mario Carneiro, 19-Apr-2013.)

Assertion

Ref	Expression
r111	⊢ 𝑅1:On–1-1→V

Proof of Theorem r111

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

Step	Hyp	Ref	Expression
1		r1fnon 9749	. . 3 ⊢ 𝑅1 Fn On
2		dffn2 6704	. . 3 ⊢ (𝑅1 Fn On ↔ 𝑅1:On⟶V)
3	1, 2	mpbi 233	. 2 ⊢ 𝑅1:On⟶V
4		eloni 6367	. . . . 5 ⊢ (𝑥 ∈ On → Ord 𝑥)
5		eloni 6367	. . . . 5 ⊢ (𝑦 ∈ On → Ord 𝑦)
6		ordtri3or 6390	. . . . 5 ⊢ ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
7	4, 5, 6	syl2an 608	. . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
8		sdomirr 9112	. . . . . . . . 9 ⊢ ¬ (𝑅1‘𝑦) ≺ (𝑅1‘𝑦)
9		r1sdom 9756	. . . . . . . . . 10 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ 𝑦) → (𝑅1‘𝑥) ≺ (𝑅1‘𝑦))
10		breq1 5106	. . . . . . . . . 10 ⊢ ((𝑅1‘𝑥) = (𝑅1‘𝑦) → ((𝑅1‘𝑥) ≺ (𝑅1‘𝑦) ↔ (𝑅1‘𝑦) ≺ (𝑅1‘𝑦)))
11	9, 10	syl5ibcom 248	. . . . . . . . 9 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ 𝑦) → ((𝑅1‘𝑥) = (𝑅1‘𝑦) → (𝑅1‘𝑦) ≺ (𝑅1‘𝑦)))
12	8, 11	mtoi 202	. . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ 𝑦) → ¬ (𝑅1‘𝑥) = (𝑅1‘𝑦))
13	12	3adant1 1148	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥 ∈ 𝑦) → ¬ (𝑅1‘𝑥) = (𝑅1‘𝑦))
14	13	pm2.21d 122	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥 ∈ 𝑦) → ((𝑅1‘𝑥) = (𝑅1‘𝑦) → 𝑥 = 𝑦))
15	14	3expia 1139	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ∈ 𝑦 → ((𝑅1‘𝑥) = (𝑅1‘𝑦) → 𝑥 = 𝑦)))
16		ax1w 13	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 = 𝑦 → ((𝑅1‘𝑥) = (𝑅1‘𝑦) → 𝑥 = 𝑦)))
17		r1sdom 9756	. . . . . . . . . 10 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (𝑅1‘𝑦) ≺ (𝑅1‘𝑥))
18		breq2 5107	. . . . . . . . . 10 ⊢ ((𝑅1‘𝑥) = (𝑅1‘𝑦) → ((𝑅1‘𝑦) ≺ (𝑅1‘𝑥) ↔ (𝑅1‘𝑦) ≺ (𝑅1‘𝑦)))
19	17, 18	syl5ibcom 248	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → ((𝑅1‘𝑥) = (𝑅1‘𝑦) → (𝑅1‘𝑦) ≺ (𝑅1‘𝑦)))
20	8, 19	mtoi 202	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → ¬ (𝑅1‘𝑥) = (𝑅1‘𝑦))
21	20	3adant2 1149	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑦 ∈ 𝑥) → ¬ (𝑅1‘𝑥) = (𝑅1‘𝑦))
22	21	pm2.21d 122	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑦 ∈ 𝑥) → ((𝑅1‘𝑥) = (𝑅1‘𝑦) → 𝑥 = 𝑦))
23	22	3expia 1139	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑦 ∈ 𝑥 → ((𝑅1‘𝑥) = (𝑅1‘𝑦) → 𝑥 = 𝑦)))
24	15, 16, 23	3jaod 1456	. . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → ((𝑅1‘𝑥) = (𝑅1‘𝑦) → 𝑥 = 𝑦)))
25	7, 24	mpd 16	. . 3 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑅1‘𝑥) = (𝑅1‘𝑦) → 𝑥 = 𝑦))
26	25	rgen2 3202	. 2 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On ((𝑅1‘𝑥) = (𝑅1‘𝑦) → 𝑥 = 𝑦)
27		dff13 7251	. 2 ⊢ (𝑅1:On–1-1→V ↔ (𝑅1:On⟶V ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On ((𝑅1‘𝑥) = (𝑅1‘𝑦) → 𝑥 = 𝑦)))
28	3, 26, 27	mpbir2an 724	1 ⊢ 𝑅1:On–1-1→V

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ w3o 1102   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3076  Vcvv 3450   class class class wbr 5103  Ord word 6356  Oncon0 6357   Fn wfn 6528  ⟶wf 6529  –1-1→wf1 6530  ‘cfv 6533   ≺ csdm 8951  𝑅₁cr1 9744

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7416  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-er 8696  df-en 8953  df-dom 8954  df-sdom 8955  df-r1 9746

This theorem is used by:  tskinf  10778  grothomex  10838  rankeq1o  36751  elhf  36754  hfninf  36766