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Theorem r111 8800
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 8792 . . 3 𝑅1 Fn On
2 dffn2 6185 . . 3 (𝑅1 Fn On ↔ 𝑅1:On⟶V)
31, 2mpbi 220 . 2 𝑅1:On⟶V
4 eloni 5874 . . . . 5 (𝑥 ∈ On → Ord 𝑥)
5 eloni 5874 . . . . 5 (𝑦 ∈ On → Ord 𝑦)
6 ordtri3or 5896 . . . . 5 ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
74, 5, 6syl2an 583 . . . 4 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
8 sdomirr 8251 . . . . . . . . 9 ¬ (𝑅1𝑦) ≺ (𝑅1𝑦)
9 r1sdom 8799 . . . . . . . . . 10 ((𝑦 ∈ On ∧ 𝑥𝑦) → (𝑅1𝑥) ≺ (𝑅1𝑦))
10 breq1 4789 . . . . . . . . . 10 ((𝑅1𝑥) = (𝑅1𝑦) → ((𝑅1𝑥) ≺ (𝑅1𝑦) ↔ (𝑅1𝑦) ≺ (𝑅1𝑦)))
119, 10syl5ibcom 235 . . . . . . . . 9 ((𝑦 ∈ On ∧ 𝑥𝑦) → ((𝑅1𝑥) = (𝑅1𝑦) → (𝑅1𝑦) ≺ (𝑅1𝑦)))
128, 11mtoi 190 . . . . . . . 8 ((𝑦 ∈ On ∧ 𝑥𝑦) → ¬ (𝑅1𝑥) = (𝑅1𝑦))
13123adant1 1124 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥𝑦) → ¬ (𝑅1𝑥) = (𝑅1𝑦))
1413pm2.21d 119 . . . . . 6 ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥𝑦) → ((𝑅1𝑥) = (𝑅1𝑦) → 𝑥 = 𝑦))
15143expia 1114 . . . . 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 8799 . . . . . . . . . 10 ((𝑥 ∈ On ∧ 𝑦𝑥) → (𝑅1𝑦) ≺ (𝑅1𝑥))
19 breq2 4790 . . . . . . . . . 10 ((𝑅1𝑥) = (𝑅1𝑦) → ((𝑅1𝑦) ≺ (𝑅1𝑥) ↔ (𝑅1𝑦) ≺ (𝑅1𝑦)))
2018, 19syl5ibcom 235 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑦𝑥) → ((𝑅1𝑥) = (𝑅1𝑦) → (𝑅1𝑦) ≺ (𝑅1𝑦)))
218, 20mtoi 190 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝑦𝑥) → ¬ (𝑅1𝑥) = (𝑅1𝑦))
22213adant2 1125 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑦𝑥) → ¬ (𝑅1𝑥) = (𝑅1𝑦))
2322pm2.21d 119 . . . . . 6 ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑦𝑥) → ((𝑅1𝑥) = (𝑅1𝑦) → 𝑥 = 𝑦))
24233expia 1114 . . . . 5 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑦𝑥 → ((𝑅1𝑥) = (𝑅1𝑦) → 𝑥 = 𝑦)))
2515, 17, 243jaod 1540 . . . 4 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑥𝑦𝑥 = 𝑦𝑦𝑥) → ((𝑅1𝑥) = (𝑅1𝑦) → 𝑥 = 𝑦)))
267, 25mpd 15 . . 3 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑅1𝑥) = (𝑅1𝑦) → 𝑥 = 𝑦))
2726rgen2a 3126 . 2 𝑥 ∈ On ∀𝑦 ∈ On ((𝑅1𝑥) = (𝑅1𝑦) → 𝑥 = 𝑦)
28 dff13 6653 . 2 (𝑅1:On–1-1→V ↔ (𝑅1:On⟶V ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On ((𝑅1𝑥) = (𝑅1𝑦) → 𝑥 = 𝑦)))
293, 27, 28mpbir2an 690 1 𝑅1:On–1-1→V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  w3o 1070  w3a 1071   = wceq 1631  wcel 2145  wral 3061  Vcvv 3351   class class class wbr 4786  Ord word 5863  Oncon0 5864   Fn wfn 6024  wf 6025  1-1wf1 6026  cfv 6029  csdm 8106  𝑅1cr1 8787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5821  df-ord 5867  df-on 5868  df-lim 5869  df-suc 5870  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-wrecs 7557  df-recs 7619  df-rdg 7657  df-er 7894  df-en 8108  df-dom 8109  df-sdom 8110  df-r1 8789
This theorem is referenced by:  tskinf  9791  grothomex  9851  rankeq1o  32608  elhf  32611  hfninf  32623
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