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Theorem nolt02olem 31969
Description: Lemma for nolt02o 31970. If 𝐴(𝑋) is undefined with 𝐴 surreal and 𝑋 ordinal, then dom 𝐴𝑋. (Contributed by Scott Fenton, 6-Dec-2021.)
Assertion
Ref Expression
nolt02olem ((𝐴 No 𝑋 ∈ On ∧ (𝐴𝑋) = ∅) → dom 𝐴𝑋)

Proof of Theorem nolt02olem
StepHypRef Expression
1 nosgnn0 31936 . . . 4 ¬ ∅ ∈ {1𝑜, 2𝑜}
21a1i 11 . . 3 ((𝐴 No 𝑋 ∈ On ∧ (𝐴𝑋) = ∅) → ¬ ∅ ∈ {1𝑜, 2𝑜})
3 simpl3 1086 . . . 4 (((𝐴 No 𝑋 ∈ On ∧ (𝐴𝑋) = ∅) ∧ 𝑋 ∈ dom 𝐴) → (𝐴𝑋) = ∅)
4 simpl1 1084 . . . . . 6 (((𝐴 No 𝑋 ∈ On ∧ (𝐴𝑋) = ∅) ∧ 𝑋 ∈ dom 𝐴) → 𝐴 No )
5 norn 31929 . . . . . 6 (𝐴 No → ran 𝐴 ⊆ {1𝑜, 2𝑜})
64, 5syl 17 . . . . 5 (((𝐴 No 𝑋 ∈ On ∧ (𝐴𝑋) = ∅) ∧ 𝑋 ∈ dom 𝐴) → ran 𝐴 ⊆ {1𝑜, 2𝑜})
7 nofun 31927 . . . . . . 7 (𝐴 No → Fun 𝐴)
873ad2ant1 1102 . . . . . 6 ((𝐴 No 𝑋 ∈ On ∧ (𝐴𝑋) = ∅) → Fun 𝐴)
9 fvelrn 6392 . . . . . 6 ((Fun 𝐴𝑋 ∈ dom 𝐴) → (𝐴𝑋) ∈ ran 𝐴)
108, 9sylan 487 . . . . 5 (((𝐴 No 𝑋 ∈ On ∧ (𝐴𝑋) = ∅) ∧ 𝑋 ∈ dom 𝐴) → (𝐴𝑋) ∈ ran 𝐴)
116, 10sseldd 3637 . . . 4 (((𝐴 No 𝑋 ∈ On ∧ (𝐴𝑋) = ∅) ∧ 𝑋 ∈ dom 𝐴) → (𝐴𝑋) ∈ {1𝑜, 2𝑜})
123, 11eqeltrrd 2731 . . 3 (((𝐴 No 𝑋 ∈ On ∧ (𝐴𝑋) = ∅) ∧ 𝑋 ∈ dom 𝐴) → ∅ ∈ {1𝑜, 2𝑜})
132, 12mtand 692 . 2 ((𝐴 No 𝑋 ∈ On ∧ (𝐴𝑋) = ∅) → ¬ 𝑋 ∈ dom 𝐴)
14 nodmon 31928 . . . 4 (𝐴 No → dom 𝐴 ∈ On)
15143ad2ant1 1102 . . 3 ((𝐴 No 𝑋 ∈ On ∧ (𝐴𝑋) = ∅) → dom 𝐴 ∈ On)
16 simp2 1082 . . 3 ((𝐴 No 𝑋 ∈ On ∧ (𝐴𝑋) = ∅) → 𝑋 ∈ On)
17 ontri1 5795 . . 3 ((dom 𝐴 ∈ On ∧ 𝑋 ∈ On) → (dom 𝐴𝑋 ↔ ¬ 𝑋 ∈ dom 𝐴))
1815, 16, 17syl2anc 694 . 2 ((𝐴 No 𝑋 ∈ On ∧ (𝐴𝑋) = ∅) → (dom 𝐴𝑋 ↔ ¬ 𝑋 ∈ dom 𝐴))
1913, 18mpbird 247 1 ((𝐴 No 𝑋 ∈ On ∧ (𝐴𝑋) = ∅) → dom 𝐴𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wss 3607  c0 3948  {cpr 4212  dom cdm 5143  ran crn 5144  Oncon0 5761  Fun wfun 5920  cfv 5926  1𝑜c1o 7598  2𝑜c2o 7599   No csur 31918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-1o 7605  df-2o 7606  df-no 31921
This theorem is referenced by:  nolt02o  31970
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