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Theorem nolt02olem 33326
 Description: Lemma for nolt02o 33327. If 𝐴(𝑋) is undefined with 𝐴 surreal and 𝑋 ordinal, then dom 𝐴 ⊆ 𝑋. (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 33293 . . . 4 ¬ ∅ ∈ {1o, 2o}
21a1i 11 . . 3 ((𝐴 No 𝑋 ∈ On ∧ (𝐴𝑋) = ∅) → ¬ ∅ ∈ {1o, 2o})
3 simpl3 1190 . . . 4 (((𝐴 No 𝑋 ∈ On ∧ (𝐴𝑋) = ∅) ∧ 𝑋 ∈ dom 𝐴) → (𝐴𝑋) = ∅)
4 simpl1 1188 . . . . . 6 (((𝐴 No 𝑋 ∈ On ∧ (𝐴𝑋) = ∅) ∧ 𝑋 ∈ dom 𝐴) → 𝐴 No )
5 norn 33286 . . . . . 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 33284 . . . . . . 7 (𝐴 No → Fun 𝐴)
873ad2ant1 1130 . . . . . 6 ((𝐴 No 𝑋 ∈ On ∧ (𝐴𝑋) = ∅) → Fun 𝐴)
9 fvelrn 6822 . . . . . 6 ((Fun 𝐴𝑋 ∈ dom 𝐴) → (𝐴𝑋) ∈ ran 𝐴)
108, 9sylan 583 . . . . 5 (((𝐴 No 𝑋 ∈ On ∧ (𝐴𝑋) = ∅) ∧ 𝑋 ∈ dom 𝐴) → (𝐴𝑋) ∈ ran 𝐴)
116, 10sseldd 3916 . . . 4 (((𝐴 No 𝑋 ∈ On ∧ (𝐴𝑋) = ∅) ∧ 𝑋 ∈ dom 𝐴) → (𝐴𝑋) ∈ {1o, 2o})
123, 11eqeltrrd 2891 . . 3 (((𝐴 No 𝑋 ∈ On ∧ (𝐴𝑋) = ∅) ∧ 𝑋 ∈ dom 𝐴) → ∅ ∈ {1o, 2o})
132, 12mtand 815 . 2 ((𝐴 No 𝑋 ∈ On ∧ (𝐴𝑋) = ∅) → ¬ 𝑋 ∈ dom 𝐴)
14 nodmon 33285 . . . 4 (𝐴 No → dom 𝐴 ∈ On)
15143ad2ant1 1130 . . 3 ((𝐴 No 𝑋 ∈ On ∧ (𝐴𝑋) = ∅) → dom 𝐴 ∈ On)
16 simp2 1134 . . 3 ((𝐴 No 𝑋 ∈ On ∧ (𝐴𝑋) = ∅) → 𝑋 ∈ On)
17 ontri1 6194 . . 3 ((dom 𝐴 ∈ On ∧ 𝑋 ∈ On) → (dom 𝐴𝑋 ↔ ¬ 𝑋 ∈ dom 𝐴))
1815, 16, 17syl2anc 587 . 2 ((𝐴 No 𝑋 ∈ On ∧ (𝐴𝑋) = ∅) → (dom 𝐴𝑋 ↔ ¬ 𝑋 ∈ dom 𝐴))
1913, 18mpbird 260 1 ((𝐴 No 𝑋 ∈ On ∧ (𝐴𝑋) = ∅) → dom 𝐴𝑋)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ⊆ wss 3881  ∅c0 4243  {cpr 4527  dom cdm 5520  ran crn 5521  Oncon0 6160  Fun wfun 6319  ‘cfv 6325  1oc1o 8081  2oc2o 8082   No csur 33275 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 5155  ax-sep 5168  ax-nul 5175  ax-pr 5296 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 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-ord 6163  df-on 6164  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-1o 8088  df-2o 8089  df-no 33278 This theorem is referenced by:  nolt02o  33327
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