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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
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