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Theorem nosepne 27659
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 27655 . . . 4 <s Or No
2 sotrine 5628 . . . 4 (( <s Or No ∧ (𝐴 No 𝐵 No )) → (𝐴𝐵 ↔ (𝐴 <s 𝐵𝐵 <s 𝐴)))
31, 2mpan 688 . . 3 ((𝐴 No 𝐵 No ) → (𝐴𝐵 ↔ (𝐴 <s 𝐵𝐵 <s 𝐴)))
4 nosepnelem 27658 . . . . 5 ((𝐴 No 𝐵 No 𝐴 <s 𝐵) → (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) ≠ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}))
543expia 1118 . . . 4 ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 → (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) ≠ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)})))
6 nosepnelem 27658 . . . . . . 7 ((𝐵 No 𝐴 No 𝐵 <s 𝐴) → (𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) ≠ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}))
7 necom 2983 . . . . . . . . . . . 12 ((𝐴𝑥) ≠ (𝐵𝑥) ↔ (𝐵𝑥) ≠ (𝐴𝑥))
87rabbii 3424 . . . . . . . . . . 11 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} = {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}
98inteqi 4954 . . . . . . . . . 10 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} = {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}
109fveq2i 6899 . . . . . . . . 9 (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)})
119fveq2i 6899 . . . . . . . . 9 (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = (𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)})
1210, 11neeq12i 2996 . . . . . . . 8 ((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) ≠ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) ↔ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) ≠ (𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}))
13 necom 2983 . . . . . . . 8 ((𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) ≠ (𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) ↔ (𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) ≠ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}))
1412, 13bitri 274 . . . . . . 7 ((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) ≠ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) ↔ (𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) ≠ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}))
156, 14sylibr 233 . . . . . 6 ((𝐵 No 𝐴 No 𝐵 <s 𝐴) → (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) ≠ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}))
16153expia 1118 . . . . 5 ((𝐵 No 𝐴 No ) → (𝐵 <s 𝐴 → (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) ≠ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)})))
1716ancoms 457 . . . 4 ((𝐴 No 𝐵 No ) → (𝐵 <s 𝐴 → (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) ≠ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)})))
185, 17jaod 857 . . 3 ((𝐴 No 𝐵 No ) → ((𝐴 <s 𝐵𝐵 <s 𝐴) → (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) ≠ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)})))
193, 18sylbid 239 . 2 ((𝐴 No 𝐵 No ) → (𝐴𝐵 → (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) ≠ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)})))
20193impia 1114 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 205  wa 394  wo 845  w3a 1084  wcel 2098  wne 2929  {crab 3418   cint 4950   class class class wbr 5149   Or wor 5589  Oncon0 6371  cfv 6549   No csur 27618   <s cslt 27619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4910  df-int 4951  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6374  df-on 6375  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fv 6557  df-1o 8487  df-2o 8488  df-no 27621  df-slt 27622
This theorem is referenced by:  nosep1o  27660  nosep2o  27661  nosepssdm  27665  noresle  27676  noetasuplem4  27715  noetainflem4  27719
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