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Theorem r111 9391
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.
StepHypRef Expression
1 r1fnon 9383 . . 3 𝑅1 Fn On
2 dffn2 6547 . . 3 (𝑅1 Fn On ↔ 𝑅1:On⟶V)
31, 2mpbi 233 . 2 𝑅1:On⟶V
4 eloni 6223 . . . . 5 (𝑥 ∈ On → Ord 𝑥)
5 eloni 6223 . . . . 5 (𝑦 ∈ On → Ord 𝑦)
6 ordtri3or 6245 . . . . 5 ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
74, 5, 6syl2an 599 . . . 4 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
8 sdomirr 8783 . . . . . . . . 9 ¬ (𝑅1𝑦) ≺ (𝑅1𝑦)
9 r1sdom 9390 . . . . . . . . . 10 ((𝑦 ∈ On ∧ 𝑥𝑦) → (𝑅1𝑥) ≺ (𝑅1𝑦))
10 breq1 5056 . . . . . . . . . 10 ((𝑅1𝑥) = (𝑅1𝑦) → ((𝑅1𝑥) ≺ (𝑅1𝑦) ↔ (𝑅1𝑦) ≺ (𝑅1𝑦)))
119, 10syl5ibcom 248 . . . . . . . . 9 ((𝑦 ∈ On ∧ 𝑥𝑦) → ((𝑅1𝑥) = (𝑅1𝑦) → (𝑅1𝑦) ≺ (𝑅1𝑦)))
128, 11mtoi 202 . . . . . . . 8 ((𝑦 ∈ On ∧ 𝑥𝑦) → ¬ (𝑅1𝑥) = (𝑅1𝑦))
13123adant1 1132 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥𝑦) → ¬ (𝑅1𝑥) = (𝑅1𝑦))
1413pm2.21d 121 . . . . . 6 ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥𝑦) → ((𝑅1𝑥) = (𝑅1𝑦) → 𝑥 = 𝑦))
15143expia 1123 . . . . 5 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥𝑦 → ((𝑅1𝑥) = (𝑅1𝑦) → 𝑥 = 𝑦)))
16 ax-1 6 . . . . . 6 (𝑥 = 𝑦 → ((𝑅1𝑥) = (𝑅1𝑦) → 𝑥 = 𝑦))
1716a1i 11 . . . . 5 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 = 𝑦 → ((𝑅1𝑥) = (𝑅1𝑦) → 𝑥 = 𝑦)))
18 r1sdom 9390 . . . . . . . . . 10 ((𝑥 ∈ On ∧ 𝑦𝑥) → (𝑅1𝑦) ≺ (𝑅1𝑥))
19 breq2 5057 . . . . . . . . . 10 ((𝑅1𝑥) = (𝑅1𝑦) → ((𝑅1𝑦) ≺ (𝑅1𝑥) ↔ (𝑅1𝑦) ≺ (𝑅1𝑦)))
2018, 19syl5ibcom 248 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑦𝑥) → ((𝑅1𝑥) = (𝑅1𝑦) → (𝑅1𝑦) ≺ (𝑅1𝑦)))
218, 20mtoi 202 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝑦𝑥) → ¬ (𝑅1𝑥) = (𝑅1𝑦))
22213adant2 1133 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑦𝑥) → ¬ (𝑅1𝑥) = (𝑅1𝑦))
2322pm2.21d 121 . . . . . 6 ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑦𝑥) → ((𝑅1𝑥) = (𝑅1𝑦) → 𝑥 = 𝑦))
24233expia 1123 . . . . 5 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑦𝑥 → ((𝑅1𝑥) = (𝑅1𝑦) → 𝑥 = 𝑦)))
2515, 17, 243jaod 1430 . . . 4 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑥𝑦𝑥 = 𝑦𝑦𝑥) → ((𝑅1𝑥) = (𝑅1𝑦) → 𝑥 = 𝑦)))
267, 25mpd 15 . . 3 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑅1𝑥) = (𝑅1𝑦) → 𝑥 = 𝑦))
2726rgen2 3124 . 2 𝑥 ∈ On ∀𝑦 ∈ On ((𝑅1𝑥) = (𝑅1𝑦) → 𝑥 = 𝑦)
28 dff13 7067 . 2 (𝑅1:On–1-1→V ↔ (𝑅1:On⟶V ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On ((𝑅1𝑥) = (𝑅1𝑦) → 𝑥 = 𝑦)))
293, 27, 28mpbir2an 711 1 𝑅1:On–1-1→V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  w3o 1088  w3a 1089   = wceq 1543  wcel 2110  wral 3061  Vcvv 3408   class class class wbr 5053  Ord word 6212  Oncon0 6213   Fn wfn 6375  wf 6376  1-1wf1 6377  cfv 6380  csdm 8625  𝑅1cr1 9378
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-er 8391  df-en 8627  df-dom 8628  df-sdom 8629  df-r1 9380
This theorem is referenced by:  tskinf  10383  grothomex  10443  rankeq1o  34210  elhf  34213  hfninf  34225
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