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Theorem nosepssdm 33889
Description: Given two non-equal surreals, their separator is less than or equal to the domain of one of them. Part of Lemma 2.1.1 of [Lipparini] p. 3. (Contributed by Scott Fenton, 6-Dec-2021.)
Assertion
Ref Expression
nosepssdm ((𝐴 No 𝐵 No 𝐴𝐵) → {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ⊆ dom 𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem nosepssdm
StepHypRef Expression
1 nosepne 33883 . . . 4 ((𝐴 No 𝐵 No 𝐴𝐵) → (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) ≠ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}))
21neneqd 2948 . . 3 ((𝐴 No 𝐵 No 𝐴𝐵) → ¬ (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}))
3 nodmord 33856 . . . . . . . . 9 (𝐴 No → Ord dom 𝐴)
433ad2ant1 1132 . . . . . . . 8 ((𝐴 No 𝐵 No 𝐴𝐵) → Ord dom 𝐴)
5 ordn2lp 6286 . . . . . . . 8 (Ord dom 𝐴 → ¬ (dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∧ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐴))
64, 5syl 17 . . . . . . 7 ((𝐴 No 𝐵 No 𝐴𝐵) → ¬ (dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∧ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐴))
7 imnan 400 . . . . . . 7 ((dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} → ¬ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐴) ↔ ¬ (dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∧ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐴))
86, 7sylibr 233 . . . . . 6 ((𝐴 No 𝐵 No 𝐴𝐵) → (dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} → ¬ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐴))
98imp 407 . . . . 5 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → ¬ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐴)
10 ndmfv 6804 . . . . 5 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐴 → (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅)
119, 10syl 17 . . . 4 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅)
12 nosepeq 33888 . . . . . 6 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → (𝐴‘dom 𝐴) = (𝐵‘dom 𝐴))
13 simpl1 1190 . . . . . . . . . . 11 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → 𝐴 No )
1413, 3syl 17 . . . . . . . . . 10 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → Ord dom 𝐴)
15 ordirr 6284 . . . . . . . . . 10 (Ord dom 𝐴 → ¬ dom 𝐴 ∈ dom 𝐴)
16 ndmfv 6804 . . . . . . . . . 10 (¬ dom 𝐴 ∈ dom 𝐴 → (𝐴‘dom 𝐴) = ∅)
1714, 15, 163syl 18 . . . . . . . . 9 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → (𝐴‘dom 𝐴) = ∅)
1817eqeq1d 2740 . . . . . . . 8 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → ((𝐴‘dom 𝐴) = (𝐵‘dom 𝐴) ↔ ∅ = (𝐵‘dom 𝐴)))
19 eqcom 2745 . . . . . . . 8 (∅ = (𝐵‘dom 𝐴) ↔ (𝐵‘dom 𝐴) = ∅)
2018, 19bitrdi 287 . . . . . . 7 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → ((𝐴‘dom 𝐴) = (𝐵‘dom 𝐴) ↔ (𝐵‘dom 𝐴) = ∅))
21 simpl2 1191 . . . . . . . . . . 11 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → 𝐵 No )
22 nofun 33852 . . . . . . . . . . 11 (𝐵 No → Fun 𝐵)
2321, 22syl 17 . . . . . . . . . 10 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → Fun 𝐵)
24 nosgnn0 33861 . . . . . . . . . . 11 ¬ ∅ ∈ {1o, 2o}
25 norn 33854 . . . . . . . . . . . . 13 (𝐵 No → ran 𝐵 ⊆ {1o, 2o})
2621, 25syl 17 . . . . . . . . . . . 12 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → ran 𝐵 ⊆ {1o, 2o})
2726sseld 3920 . . . . . . . . . . 11 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → (∅ ∈ ran 𝐵 → ∅ ∈ {1o, 2o}))
2824, 27mtoi 198 . . . . . . . . . 10 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → ¬ ∅ ∈ ran 𝐵)
29 funeldmb 33737 . . . . . . . . . 10 ((Fun 𝐵 ∧ ¬ ∅ ∈ ran 𝐵) → (dom 𝐴 ∈ dom 𝐵 ↔ (𝐵‘dom 𝐴) ≠ ∅))
3023, 28, 29syl2anc 584 . . . . . . . . 9 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → (dom 𝐴 ∈ dom 𝐵 ↔ (𝐵‘dom 𝐴) ≠ ∅))
3130necon2bbid 2987 . . . . . . . 8 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → ((𝐵‘dom 𝐴) = ∅ ↔ ¬ dom 𝐴 ∈ dom 𝐵))
32 nodmord 33856 . . . . . . . . . . . 12 (𝐵 No → Ord dom 𝐵)
33323ad2ant2 1133 . . . . . . . . . . 11 ((𝐴 No 𝐵 No 𝐴𝐵) → Ord dom 𝐵)
34 ordtr1 6309 . . . . . . . . . . 11 (Ord dom 𝐵 → ((dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∧ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐵) → dom 𝐴 ∈ dom 𝐵))
3533, 34syl 17 . . . . . . . . . 10 ((𝐴 No 𝐵 No 𝐴𝐵) → ((dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∧ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐵) → dom 𝐴 ∈ dom 𝐵))
3635expdimp 453 . . . . . . . . 9 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → ( {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐵 → dom 𝐴 ∈ dom 𝐵))
3736con3d 152 . . . . . . . 8 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → (¬ dom 𝐴 ∈ dom 𝐵 → ¬ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐵))
3831, 37sylbid 239 . . . . . . 7 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → ((𝐵‘dom 𝐴) = ∅ → ¬ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐵))
3920, 38sylbid 239 . . . . . 6 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → ((𝐴‘dom 𝐴) = (𝐵‘dom 𝐴) → ¬ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐵))
4012, 39mpd 15 . . . . 5 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → ¬ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐵)
41 ndmfv 6804 . . . . 5 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐵 → (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅)
4240, 41syl 17 . . . 4 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅)
4311, 42eqtr4d 2781 . . 3 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}))
442, 43mtand 813 . 2 ((𝐴 No 𝐵 No 𝐴𝐵) → ¬ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)})
45 nosepon 33868 . . 3 ((𝐴 No 𝐵 No 𝐴𝐵) → {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ On)
46 nodmon 33853 . . . 4 (𝐴 No → dom 𝐴 ∈ On)
47463ad2ant1 1132 . . 3 ((𝐴 No 𝐵 No 𝐴𝐵) → dom 𝐴 ∈ On)
48 ontri1 6300 . . 3 (( {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ On ∧ dom 𝐴 ∈ On) → ( {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ⊆ dom 𝐴 ↔ ¬ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}))
4945, 47, 48syl2anc 584 . 2 ((𝐴 No 𝐵 No 𝐴𝐵) → ( {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ⊆ dom 𝐴 ↔ ¬ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}))
5044, 49mpbird 256 1 ((𝐴 No 𝐵 No 𝐴𝐵) → {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ⊆ dom 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1086   = wceq 1539  wcel 2106  wne 2943  {crab 3068  wss 3887  c0 4256  {cpr 4563   cint 4879  dom cdm 5589  ran crn 5590  Ord word 6265  Oncon0 6266  Fun wfun 6427  cfv 6433  1oc1o 8290  2oc2o 8291   No csur 33843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-1o 8297  df-2o 8298  df-no 33846  df-slt 33847
This theorem is referenced by:  nosupbnd2lem1  33918  noinfbnd2lem1  33933  noetasuplem4  33939  noetainflem4  33943
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