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Theorem nosepne 33082
Description: The value of two non-equal surreals at the first place they differ is different. (Contributed by Scott Fenton, 24-Nov-2021.)
Assertion
Ref Expression
nosepne ((𝐴 No 𝐵 No 𝐴𝐵) → (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) ≠ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem nosepne
StepHypRef Expression
1 sltso 33078 . . . 4 <s Or No
2 sotrine 32900 . . . 4 (( <s Or No ∧ (𝐴 No 𝐵 No )) → (𝐴𝐵 ↔ (𝐴 <s 𝐵𝐵 <s 𝐴)))
31, 2mpan 686 . . 3 ((𝐴 No 𝐵 No ) → (𝐴𝐵 ↔ (𝐴 <s 𝐵𝐵 <s 𝐴)))
4 nosepnelem 33081 . . . . 5 ((𝐴 No 𝐵 No 𝐴 <s 𝐵) → (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) ≠ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}))
543expia 1113 . . . 4 ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 → (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) ≠ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)})))
6 nosepnelem 33081 . . . . . . 7 ((𝐵 No 𝐴 No 𝐵 <s 𝐴) → (𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) ≠ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}))
7 necom 3066 . . . . . . . . . . . 12 ((𝐴𝑥) ≠ (𝐵𝑥) ↔ (𝐵𝑥) ≠ (𝐴𝑥))
87rabbii 3471 . . . . . . . . . . 11 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} = {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}
98inteqi 4871 . . . . . . . . . 10 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} = {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}
109fveq2i 6666 . . . . . . . . 9 (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)})
119fveq2i 6666 . . . . . . . . 9 (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = (𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)})
1210, 11neeq12i 3079 . . . . . . . 8 ((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) ≠ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) ↔ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) ≠ (𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}))
13 necom 3066 . . . . . . . 8 ((𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) ≠ (𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) ↔ (𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) ≠ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}))
1412, 13bitri 276 . . . . . . 7 ((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) ≠ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) ↔ (𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) ≠ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}))
156, 14sylibr 235 . . . . . 6 ((𝐵 No 𝐴 No 𝐵 <s 𝐴) → (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) ≠ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}))
16153expia 1113 . . . . 5 ((𝐵 No 𝐴 No ) → (𝐵 <s 𝐴 → (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) ≠ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)})))
1716ancoms 459 . . . 4 ((𝐴 No 𝐵 No ) → (𝐵 <s 𝐴 → (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) ≠ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)})))
185, 17jaod 853 . . 3 ((𝐴 No 𝐵 No ) → ((𝐴 <s 𝐵𝐵 <s 𝐴) → (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) ≠ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)})))
193, 18sylbid 241 . 2 ((𝐴 No 𝐵 No ) → (𝐴𝐵 → (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) ≠ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)})))
20193impia 1109 1 ((𝐴 No 𝐵 No 𝐴𝐵) → (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) ≠ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wo 841  w3a 1079  wcel 2105  wne 3013  {crab 3139   cint 4867   class class class wbr 5057   Or wor 5466  Oncon0 6184  cfv 6348   No csur 33044   <s cslt 33045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-1o 8091  df-2o 8092  df-no 33047  df-slt 33048
This theorem is referenced by:  nosep1o  33083  nosepssdm  33087  noresle  33097  noetalem3  33116
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