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Theorem nosepdm 33624
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 33616 . . . 4 <s Or No
2 sotrine 33453 . . . 4 (( <s Or No ∧ (𝐴 No 𝐵 No )) → (𝐴𝐵 ↔ (𝐴 <s 𝐵𝐵 <s 𝐴)))
31, 2mpan 690 . . 3 ((𝐴 No 𝐵 No ) → (𝐴𝐵 ↔ (𝐴 <s 𝐵𝐵 <s 𝐴)))
4 nosepdmlem 33623 . . . . . 6 ((𝐴 No 𝐵 No 𝐴 <s 𝐵) → {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ (dom 𝐴 ∪ dom 𝐵))
543expa 1120 . . . . 5 (((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) → {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ (dom 𝐴 ∪ dom 𝐵))
6 simplr 769 . . . . . . 7 (((𝐴 No 𝐵 No ) ∧ 𝐵 <s 𝐴) → 𝐵 No )
7 simpll 767 . . . . . . 7 (((𝐴 No 𝐵 No ) ∧ 𝐵 <s 𝐴) → 𝐴 No )
8 simpr 488 . . . . . . 7 (((𝐴 No 𝐵 No ) ∧ 𝐵 <s 𝐴) → 𝐵 <s 𝐴)
9 nosepdmlem 33623 . . . . . . 7 ((𝐵 No 𝐴 No 𝐵 <s 𝐴) → {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)} ∈ (dom 𝐵 ∪ dom 𝐴))
106, 7, 8, 9syl3anc 1373 . . . . . 6 (((𝐴 No 𝐵 No ) ∧ 𝐵 <s 𝐴) → {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)} ∈ (dom 𝐵 ∪ dom 𝐴))
11 necom 2994 . . . . . . . 8 ((𝐴𝑥) ≠ (𝐵𝑥) ↔ (𝐵𝑥) ≠ (𝐴𝑥))
1211rabbii 3383 . . . . . . 7 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} = {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}
1312inteqi 4863 . . . . . 6 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} = {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}
14 uncom 4067 . . . . . 6 (dom 𝐴 ∪ dom 𝐵) = (dom 𝐵 ∪ dom 𝐴)
1510, 13, 143eltr4g 2855 . . . . 5 (((𝐴 No 𝐵 No ) ∧ 𝐵 <s 𝐴) → {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ (dom 𝐴 ∪ dom 𝐵))
165, 15jaodan 958 . . . 4 (((𝐴 No 𝐵 No ) ∧ (𝐴 <s 𝐵𝐵 <s 𝐴)) → {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ (dom 𝐴 ∪ dom 𝐵))
1716ex 416 . . 3 ((𝐴 No 𝐵 No ) → ((𝐴 <s 𝐵𝐵 <s 𝐴) → {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ (dom 𝐴 ∪ dom 𝐵)))
183, 17sylbid 243 . 2 ((𝐴 No 𝐵 No ) → (𝐴𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ (dom 𝐴 ∪ dom 𝐵)))
19183impia 1119 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 209  wa 399  wo 847  w3a 1089  wcel 2110  wne 2940  {crab 3065  cun 3864   cint 4859   class class class wbr 5053   Or wor 5467  dom cdm 5551  Oncon0 6213  cfv 6380   No csur 33580   <s cslt 33581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-ord 6216  df-on 6217  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-fv 6388  df-1o 8202  df-2o 8203  df-no 33583  df-slt 33584
This theorem is referenced by:  nodenselem5  33628  noresle  33637
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