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Theorem nosepeq 32373
Description: The values of two surreals at a point less than their separators are equal. (Contributed by Scott Fenton, 6-Dec-2021.)
Assertion
Ref Expression
nosepeq (((𝐴 No 𝐵 No 𝐴𝐵) ∧ 𝑋 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → (𝐴𝑋) = (𝐵𝑋))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑋

Proof of Theorem nosepeq
StepHypRef Expression
1 nosepon 32356 . . . 4 ((𝐴 No 𝐵 No 𝐴𝐵) → {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ On)
2 onelon 5987 . . . 4 (( {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ On ∧ 𝑋 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → 𝑋 ∈ On)
31, 2sylan 577 . . 3 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ 𝑋 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → 𝑋 ∈ On)
4 simpr 479 . . 3 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ 𝑋 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → 𝑋 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)})
5 fveq2 6432 . . . . 5 (𝑥 = 𝑋 → (𝐴𝑥) = (𝐴𝑋))
6 fveq2 6432 . . . . 5 (𝑥 = 𝑋 → (𝐵𝑥) = (𝐵𝑋))
75, 6neeq12d 3059 . . . 4 (𝑥 = 𝑋 → ((𝐴𝑥) ≠ (𝐵𝑥) ↔ (𝐴𝑋) ≠ (𝐵𝑋)))
87onnminsb 7264 . . 3 (𝑋 ∈ On → (𝑋 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} → ¬ (𝐴𝑋) ≠ (𝐵𝑋)))
93, 4, 8sylc 65 . 2 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ 𝑋 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → ¬ (𝐴𝑋) ≠ (𝐵𝑋))
10 df-ne 2999 . . 3 ((𝐴𝑋) ≠ (𝐵𝑋) ↔ ¬ (𝐴𝑋) = (𝐵𝑋))
1110con2bii 349 . 2 ((𝐴𝑋) = (𝐵𝑋) ↔ ¬ (𝐴𝑋) ≠ (𝐵𝑋))
129, 11sylibr 226 1 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ 𝑋 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → (𝐴𝑋) = (𝐵𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 386  w3a 1113   = wceq 1658  wcel 2166  wne 2998  {crab 3120   cint 4696  Oncon0 5962  cfv 6122   No csur 32331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-rep 4993  ax-sep 5004  ax-nul 5012  ax-pow 5064  ax-pr 5126  ax-un 7208
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ne 2999  df-ral 3121  df-rex 3122  df-reu 3123  df-rab 3125  df-v 3415  df-sbc 3662  df-csb 3757  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-pss 3813  df-nul 4144  df-if 4306  df-sn 4397  df-pr 4399  df-tp 4401  df-op 4403  df-uni 4658  df-int 4697  df-iun 4741  df-br 4873  df-opab 4935  df-mpt 4952  df-tr 4975  df-id 5249  df-eprel 5254  df-po 5262  df-so 5263  df-fr 5300  df-we 5302  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-ima 5354  df-ord 5965  df-on 5966  df-suc 5968  df-iota 6085  df-fun 6124  df-fn 6125  df-f 6126  df-f1 6127  df-fo 6128  df-f1o 6129  df-fv 6130  df-1o 7825  df-2o 7826  df-no 32334
This theorem is referenced by:  nosepssdm  32374  nodenselem7  32378
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