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Theorem nosepeq 32141
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 32124 . . . 4 ((𝐴 No 𝐵 No 𝐴𝐵) → {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ On)
2 onelon 5909 . . . 4 (( {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ On ∧ 𝑋 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → 𝑋 ∈ On)
31, 2sylan 489 . . 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 6352 . . . . 5 (𝑥 = 𝑋 → (𝐴𝑥) = (𝐴𝑋))
6 fveq2 6352 . . . . 5 (𝑥 = 𝑋 → (𝐵𝑥) = (𝐵𝑋))
75, 6neeq12d 2993 . . . 4 (𝑥 = 𝑋 → ((𝐴𝑥) ≠ (𝐵𝑥) ↔ (𝐴𝑋) ≠ (𝐵𝑋)))
87onnminsb 7169 . . 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 2933 . . 3 ((𝐴𝑋) ≠ (𝐵𝑋) ↔ ¬ (𝐴𝑋) = (𝐵𝑋))
1110con2bii 346 . 2 ((𝐴𝑋) = (𝐵𝑋) ↔ ¬ (𝐴𝑋) ≠ (𝐵𝑋))
129, 11sylibr 224 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 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932  {crab 3054   cint 4627  Oncon0 5884  cfv 6049   No csur 32099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-ord 5887  df-on 5888  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-1o 7729  df-2o 7730  df-no 32102
This theorem is referenced by:  nosepssdm  32142  nodenselem7  32146
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