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Theorem nosepssdm 32173
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 32168 . . . 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 32143 . . . . . . . . 9 (𝐴 No → Ord dom 𝐴)
433ad2ant1 1127 . . . . . . . 8 ((𝐴 No 𝐵 No 𝐴𝐵) → Ord dom 𝐴)
5 ordn2lp 5885 . . . . . . . 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 386 . . . . . . 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 224 . . . . . 6 ((𝐴 No 𝐵 No 𝐴𝐵) → (dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} → ¬ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐴))
98imp 393 . . . . 5 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → ¬ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐴)
10 ndmfv 6361 . . . . 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 32172 . . . . . 6 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → (𝐴‘dom 𝐴) = (𝐵‘dom 𝐴))
13 simpl1 1227 . . . . . . . . . . 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 5883 . . . . . . . . . 10 (Ord dom 𝐴 → ¬ dom 𝐴 ∈ dom 𝐴)
16 ndmfv 6361 . . . . . . . . . 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 2773 . . . . . . . 8 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → ((𝐴‘dom 𝐴) = (𝐵‘dom 𝐴) ↔ ∅ = (𝐵‘dom 𝐴)))
19 eqcom 2778 . . . . . . . 8 (∅ = (𝐵‘dom 𝐴) ↔ (𝐵‘dom 𝐴) = ∅)
2018, 19syl6bb 276 . . . . . . 7 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → ((𝐴‘dom 𝐴) = (𝐵‘dom 𝐴) ↔ (𝐵‘dom 𝐴) = ∅))
21 simpl2 1229 . . . . . . . . . . 11 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → 𝐵 No )
22 nofun 32139 . . . . . . . . . . 11 (𝐵 No → Fun 𝐵)
2321, 22syl 17 . . . . . . . . . 10 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → Fun 𝐵)
24 nosgnn0 32148 . . . . . . . . . . 11 ¬ ∅ ∈ {1𝑜, 2𝑜}
25 norn 32141 . . . . . . . . . . . . 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 3751 . . . . . . . . . . 11 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → (∅ ∈ ran 𝐵 → ∅ ∈ {1𝑜, 2𝑜}))
2824, 27mtoi 190 . . . . . . . . . 10 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → ¬ ∅ ∈ ran 𝐵)
29 funeldmb 31999 . . . . . . . . . 10 ((Fun 𝐵 ∧ ¬ ∅ ∈ ran 𝐵) → (dom 𝐴 ∈ dom 𝐵 ↔ (𝐵‘dom 𝐴) ≠ ∅))
3023, 28, 29syl2anc 573 . . . . . . . . 9 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → (dom 𝐴 ∈ dom 𝐵 ↔ (𝐵‘dom 𝐴) ≠ ∅))
3130necon2bbid 2986 . . . . . . . 8 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → ((𝐵‘dom 𝐴) = ∅ ↔ ¬ dom 𝐴 ∈ dom 𝐵))
32 nodmord 32143 . . . . . . . . . . . 12 (𝐵 No → Ord dom 𝐵)
33323ad2ant2 1128 . . . . . . . . . . 11 ((𝐴 No 𝐵 No 𝐴𝐵) → Ord dom 𝐵)
34 ordtr1 5909 . . . . . . . . . . 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 440 . . . . . . . . 9 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → ( {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐵 → dom 𝐴 ∈ dom 𝐵))
3736con3d 149 . . . . . . . 8 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → (¬ dom 𝐴 ∈ dom 𝐵 → ¬ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐵))
3831, 37sylbid 230 . . . . . . 7 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → ((𝐵‘dom 𝐴) = ∅ → ¬ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐵))
3920, 38sylbid 230 . . . . . 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 6361 . . . . 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 2808 . . 3 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}))
442, 43mtand 817 . 2 ((𝐴 No 𝐵 No 𝐴𝐵) → ¬ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)})
45 nosepon 32155 . . 3 ((𝐴 No 𝐵 No 𝐴𝐵) → {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ On)
46 nodmon 32140 . . . 4 (𝐴 No → dom 𝐴 ∈ On)
47463ad2ant1 1127 . . 3 ((𝐴 No 𝐵 No 𝐴𝐵) → dom 𝐴 ∈ On)
48 ontri1 5899 . . 3 (( {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ On ∧ dom 𝐴 ∈ On) → ( {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ⊆ dom 𝐴 ↔ ¬ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}))
4945, 47, 48syl2anc 573 . 2 ((𝐴 No 𝐵 No 𝐴𝐵) → ( {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ⊆ dom 𝐴 ↔ ¬ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}))
5044, 49mpbird 247 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 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wne 2943  {crab 3065  wss 3723  c0 4063  {cpr 4319   cint 4612  dom cdm 5250  ran crn 5251  Ord word 5864  Oncon0 5865  Fun wfun 6024  cfv 6030  1𝑜c1o 7710  2𝑜c2o 7711   No csur 32130
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 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
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 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-ord 5868  df-on 5869  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-1o 7717  df-2o 7718  df-no 32133  df-slt 32134
This theorem is referenced by:  nosupbnd2lem1  32198  noetalem3  32202
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