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Theorem r111 9206
 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 9198 . . 3 𝑅1 Fn On
2 dffn2 6497 . . 3 (𝑅1 Fn On ↔ 𝑅1:On⟶V)
31, 2mpbi 233 . 2 𝑅1:On⟶V
4 eloni 6176 . . . . 5 (𝑥 ∈ On → Ord 𝑥)
5 eloni 6176 . . . . 5 (𝑦 ∈ On → Ord 𝑦)
6 ordtri3or 6198 . . . . 5 ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
74, 5, 6syl2an 598 . . . 4 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
8 sdomirr 8656 . . . . . . . . 9 ¬ (𝑅1𝑦) ≺ (𝑅1𝑦)
9 r1sdom 9205 . . . . . . . . . 10 ((𝑦 ∈ On ∧ 𝑥𝑦) → (𝑅1𝑥) ≺ (𝑅1𝑦))
10 breq1 5037 . . . . . . . . . 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 1127 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥𝑦) → ¬ (𝑅1𝑥) = (𝑅1𝑦))
1413pm2.21d 121 . . . . . 6 ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥𝑦) → ((𝑅1𝑥) = (𝑅1𝑦) → 𝑥 = 𝑦))
15143expia 1118 . . . . 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 9205 . . . . . . . . . 10 ((𝑥 ∈ On ∧ 𝑦𝑥) → (𝑅1𝑦) ≺ (𝑅1𝑥))
19 breq2 5038 . . . . . . . . . 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 1128 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑦𝑥) → ¬ (𝑅1𝑥) = (𝑅1𝑦))
2322pm2.21d 121 . . . . . 6 ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑦𝑥) → ((𝑅1𝑥) = (𝑅1𝑦) → 𝑥 = 𝑦))
24233expia 1118 . . . . 5 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑦𝑥 → ((𝑅1𝑥) = (𝑅1𝑦) → 𝑥 = 𝑦)))
2515, 17, 243jaod 1425 . . . 4 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑥𝑦𝑥 = 𝑦𝑦𝑥) → ((𝑅1𝑥) = (𝑅1𝑦) → 𝑥 = 𝑦)))
267, 25mpd 15 . . 3 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑅1𝑥) = (𝑅1𝑦) → 𝑥 = 𝑦))
2726rgen2 3168 . 2 𝑥 ∈ On ∀𝑦 ∈ On ((𝑅1𝑥) = (𝑅1𝑦) → 𝑥 = 𝑦)
28 dff13 7001 . 2 (𝑅1:On–1-1→V ↔ (𝑅1:On⟶V ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On ((𝑅1𝑥) = (𝑅1𝑦) → 𝑥 = 𝑦)))
293, 27, 28mpbir2an 710 1 𝑅1:On–1-1→V
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ w3o 1083   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  Vcvv 3442   class class class wbr 5034  Ord word 6165  Oncon0 6166   Fn wfn 6327  ⟶wf 6328  –1-1→wf1 6329  ‘cfv 6332   ≺ csdm 8509  𝑅1cr1 9193 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4805  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-tr 5141  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-wrecs 7948  df-recs 8009  df-rdg 8047  df-er 8290  df-en 8511  df-dom 8512  df-sdom 8513  df-r1 9195 This theorem is referenced by:  tskinf  10198  grothomex  10258  rankeq1o  33892  elhf  33895  hfninf  33907
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