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Theorem nosepssdm 31961
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 31956 . . . 4 ((𝐴 No 𝐵 No 𝐴𝐵) → (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) ≠ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}))
21neneqd 2828 . . 3 ((𝐴 No 𝐵 No 𝐴𝐵) → ¬ (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}))
3 nodmord 31931 . . . . . . . . 9 (𝐴 No → Ord dom 𝐴)
433ad2ant1 1102 . . . . . . . 8 ((𝐴 No 𝐵 No 𝐴𝐵) → Ord dom 𝐴)
5 ordn2lp 5781 . . . . . . . 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 437 . . . . . . 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 444 . . . . 5 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → ¬ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐴)
10 ndmfv 6256 . . . . 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 31960 . . . . . 6 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → (𝐴‘dom 𝐴) = (𝐵‘dom 𝐴))
13 simpl1 1084 . . . . . . . . . . 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 5779 . . . . . . . . . 10 (Ord dom 𝐴 → ¬ dom 𝐴 ∈ dom 𝐴)
16 ndmfv 6256 . . . . . . . . . 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 2653 . . . . . . . 8 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → ((𝐴‘dom 𝐴) = (𝐵‘dom 𝐴) ↔ ∅ = (𝐵‘dom 𝐴)))
19 eqcom 2658 . . . . . . . 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 1085 . . . . . . . . . . 11 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → 𝐵 No )
22 nofun 31927 . . . . . . . . . . 11 (𝐵 No → Fun 𝐵)
2321, 22syl 17 . . . . . . . . . 10 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → Fun 𝐵)
24 nosgnn0 31936 . . . . . . . . . . 11 ¬ ∅ ∈ {1𝑜, 2𝑜}
25 norn 31929 . . . . . . . . . . . . 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 3635 . . . . . . . . . . 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 31787 . . . . . . . . . 10 ((Fun 𝐵 ∧ ¬ ∅ ∈ ran 𝐵) → (dom 𝐴 ∈ dom 𝐵 ↔ (𝐵‘dom 𝐴) ≠ ∅))
3023, 28, 29syl2anc 694 . . . . . . . . 9 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → (dom 𝐴 ∈ dom 𝐵 ↔ (𝐵‘dom 𝐴) ≠ ∅))
3130necon2bbid 2866 . . . . . . . 8 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → ((𝐵‘dom 𝐴) = ∅ ↔ ¬ dom 𝐴 ∈ dom 𝐵))
32 nodmord 31931 . . . . . . . . . . . 12 (𝐵 No → Ord dom 𝐵)
33323ad2ant2 1103 . . . . . . . . . . 11 ((𝐴 No 𝐵 No 𝐴𝐵) → Ord dom 𝐵)
34 ordtr1 5805 . . . . . . . . . . 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 452 . . . . . . . . 9 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → ( {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐵 → dom 𝐴 ∈ dom 𝐵))
3736con3d 148 . . . . . . . 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 6256 . . . . 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 2688 . . 3 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}))
442, 43mtand 692 . 2 ((𝐴 No 𝐵 No 𝐴𝐵) → ¬ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)})
45 nosepon 31943 . . 3 ((𝐴 No 𝐵 No 𝐴𝐵) → {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ On)
46 nodmon 31928 . . . 4 (𝐴 No → dom 𝐴 ∈ On)
47463ad2ant1 1102 . . 3 ((𝐴 No 𝐵 No 𝐴𝐵) → dom 𝐴 ∈ On)
48 ontri1 5795 . . 3 (( {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ On ∧ dom 𝐴 ∈ On) → ( {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ⊆ dom 𝐴 ↔ ¬ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}))
4945, 47, 48syl2anc 694 . 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 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  {crab 2945  wss 3607  c0 3948  {cpr 4212   cint 4507  dom cdm 5143  ran crn 5144  Ord word 5760  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-8 2032  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-pow 4873  ax-pr 4936  ax-un 6991
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-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  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  df-slt 31922
This theorem is referenced by:  nosupbnd2lem1  31986  noetalem3  31990
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