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Theorem nosepdm 31959
Description: The first place two surreals differ is an element of the larger of their domains. (Contributed by Scott Fenton, 24-Nov-2021.)
Assertion
Ref Expression
nosepdm ((𝐴 No 𝐵 No 𝐴𝐵) → {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ (dom 𝐴 ∪ dom 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem nosepdm
StepHypRef Expression
1 sltso 31952 . . . 4 <s Or No
2 sotrine 31784 . . . 4 (( <s Or No ∧ (𝐴 No 𝐵 No )) → (𝐴𝐵 ↔ (𝐴 <s 𝐵𝐵 <s 𝐴)))
31, 2mpan 706 . . 3 ((𝐴 No 𝐵 No ) → (𝐴𝐵 ↔ (𝐴 <s 𝐵𝐵 <s 𝐴)))
4 nosepdmlem 31958 . . . . . 6 ((𝐴 No 𝐵 No 𝐴 <s 𝐵) → {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ (dom 𝐴 ∪ dom 𝐵))
543expa 1284 . . . . 5 (((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) → {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ (dom 𝐴 ∪ dom 𝐵))
6 simplr 807 . . . . . . 7 (((𝐴 No 𝐵 No ) ∧ 𝐵 <s 𝐴) → 𝐵 No )
7 simpll 805 . . . . . . 7 (((𝐴 No 𝐵 No ) ∧ 𝐵 <s 𝐴) → 𝐴 No )
8 simpr 476 . . . . . . 7 (((𝐴 No 𝐵 No ) ∧ 𝐵 <s 𝐴) → 𝐵 <s 𝐴)
9 nosepdmlem 31958 . . . . . . 7 ((𝐵 No 𝐴 No 𝐵 <s 𝐴) → {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)} ∈ (dom 𝐵 ∪ dom 𝐴))
106, 7, 8, 9syl3anc 1366 . . . . . 6 (((𝐴 No 𝐵 No ) ∧ 𝐵 <s 𝐴) → {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)} ∈ (dom 𝐵 ∪ dom 𝐴))
11 necom 2876 . . . . . . . 8 ((𝐴𝑥) ≠ (𝐵𝑥) ↔ (𝐵𝑥) ≠ (𝐴𝑥))
1211rabbii 3216 . . . . . . 7 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} = {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}
1312inteqi 4511 . . . . . 6 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} = {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}
14 uncom 3790 . . . . . 6 (dom 𝐴 ∪ dom 𝐵) = (dom 𝐵 ∪ dom 𝐴)
1510, 13, 143eltr4g 2747 . . . . 5 (((𝐴 No 𝐵 No ) ∧ 𝐵 <s 𝐴) → {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ (dom 𝐴 ∪ dom 𝐵))
165, 15jaodan 843 . . . 4 (((𝐴 No 𝐵 No ) ∧ (𝐴 <s 𝐵𝐵 <s 𝐴)) → {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ (dom 𝐴 ∪ dom 𝐵))
1716ex 449 . . 3 ((𝐴 No 𝐵 No ) → ((𝐴 <s 𝐵𝐵 <s 𝐴) → {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ (dom 𝐴 ∪ dom 𝐵)))
183, 17sylbid 230 . 2 ((𝐴 No 𝐵 No ) → (𝐴𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ (dom 𝐴 ∪ dom 𝐵)))
19183impia 1280 1 ((𝐴 No 𝐵 No 𝐴𝐵) → {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ (dom 𝐴 ∪ dom 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383  w3a 1054  wcel 2030  wne 2823  {crab 2945  cun 3605   cint 4507   class class class wbr 4685   Or wor 5063  dom cdm 5143  Oncon0 5761  cfv 5926   No csur 31918   <s cslt 31919
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-1o 7605  df-2o 7606  df-no 31921  df-slt 31922
This theorem is referenced by:  nodenselem5  31963  noresle  31971
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