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Theorem nosepssdm 27057
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 27051 . . . 4 ((𝐴 No 𝐵 No 𝐴𝐵) → (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) ≠ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}))
21neneqd 2945 . . 3 ((𝐴 No 𝐵 No 𝐴𝐵) → ¬ (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}))
3 nodmord 27024 . . . . . . . . 9 (𝐴 No → Ord dom 𝐴)
433ad2ant1 1134 . . . . . . . 8 ((𝐴 No 𝐵 No 𝐴𝐵) → Ord dom 𝐴)
5 ordn2lp 6341 . . . . . . . 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 401 . . . . . . 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 408 . . . . 5 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → ¬ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐴)
10 ndmfv 6881 . . . . 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 27056 . . . . . 6 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → (𝐴‘dom 𝐴) = (𝐵‘dom 𝐴))
13 simpl1 1192 . . . . . . . . . . 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 6339 . . . . . . . . . 10 (Ord dom 𝐴 → ¬ dom 𝐴 ∈ dom 𝐴)
16 ndmfv 6881 . . . . . . . . . 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 2735 . . . . . . . 8 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → ((𝐴‘dom 𝐴) = (𝐵‘dom 𝐴) ↔ ∅ = (𝐵‘dom 𝐴)))
19 eqcom 2740 . . . . . . . 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 1193 . . . . . . . . . . 11 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → 𝐵 No )
22 nofun 27020 . . . . . . . . . . 11 (𝐵 No → Fun 𝐵)
2321, 22syl 17 . . . . . . . . . 10 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → Fun 𝐵)
24 nosgnn0 27029 . . . . . . . . . . 11 ¬ ∅ ∈ {1o, 2o}
25 norn 27022 . . . . . . . . . . . . 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 3947 . . . . . . . . . . 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 7308 . . . . . . . . . 10 ((Fun 𝐵 ∧ ¬ ∅ ∈ ran 𝐵) → (dom 𝐴 ∈ dom 𝐵 ↔ (𝐵‘dom 𝐴) ≠ ∅))
3023, 28, 29syl2anc 585 . . . . . . . . 9 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → (dom 𝐴 ∈ dom 𝐵 ↔ (𝐵‘dom 𝐴) ≠ ∅))
3130necon2bbid 2984 . . . . . . . 8 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → ((𝐵‘dom 𝐴) = ∅ ↔ ¬ dom 𝐴 ∈ dom 𝐵))
32 nodmord 27024 . . . . . . . . . . . 12 (𝐵 No → Ord dom 𝐵)
33323ad2ant2 1135 . . . . . . . . . . 11 ((𝐴 No 𝐵 No 𝐴𝐵) → Ord dom 𝐵)
34 ordtr1 6364 . . . . . . . . . . 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 454 . . . . . . . . 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 6881 . . . . 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 2776 . . 3 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}))
442, 43mtand 815 . 2 ((𝐴 No 𝐵 No 𝐴𝐵) → ¬ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)})
45 nosepon 27036 . . 3 ((𝐴 No 𝐵 No 𝐴𝐵) → {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ On)
46 nodmon 27021 . . . 4 (𝐴 No → dom 𝐴 ∈ On)
47463ad2ant1 1134 . . 3 ((𝐴 No 𝐵 No 𝐴𝐵) → dom 𝐴 ∈ On)
48 ontri1 6355 . . 3 (( {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ On ∧ dom 𝐴 ∈ On) → ( {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ⊆ dom 𝐴 ↔ ¬ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}))
4945, 47, 48syl2anc 585 . 2 ((𝐴 No 𝐵 No 𝐴𝐵) → ( {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ⊆ dom 𝐴 ↔ ¬ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}))
5044, 49mpbird 257 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 397  w3a 1088   = wceq 1542  wcel 2107  wne 2940  {crab 3406  wss 3914  c0 4286  {cpr 4592   cint 4911  dom cdm 5637  ran crn 5638  Ord word 6320  Oncon0 6321  Fun wfun 6494  cfv 6500  1oc1o 8409  2oc2o 8410   No csur 27011
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ord 6324  df-on 6325  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-1o 8416  df-2o 8417  df-no 27014  df-slt 27015
This theorem is referenced by:  nosupbnd2lem1  27086  noinfbnd2lem1  27101  noetasuplem4  27107  noetainflem4  27111
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