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Theorem nosepdm 32791
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 32784 . . . 4 <s Or No
2 sotrine 32605 . . . 4 (( <s Or No ∧ (𝐴 No 𝐵 No )) → (𝐴𝐵 ↔ (𝐴 <s 𝐵𝐵 <s 𝐴)))
31, 2mpan 686 . . 3 ((𝐴 No 𝐵 No ) → (𝐴𝐵 ↔ (𝐴 <s 𝐵𝐵 <s 𝐴)))
4 nosepdmlem 32790 . . . . . 6 ((𝐴 No 𝐵 No 𝐴 <s 𝐵) → {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ (dom 𝐴 ∪ dom 𝐵))
543expa 1111 . . . . 5 (((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) → {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ (dom 𝐴 ∪ dom 𝐵))
6 simplr 765 . . . . . . 7 (((𝐴 No 𝐵 No ) ∧ 𝐵 <s 𝐴) → 𝐵 No )
7 simpll 763 . . . . . . 7 (((𝐴 No 𝐵 No ) ∧ 𝐵 <s 𝐴) → 𝐴 No )
8 simpr 485 . . . . . . 7 (((𝐴 No 𝐵 No ) ∧ 𝐵 <s 𝐴) → 𝐵 <s 𝐴)
9 nosepdmlem 32790 . . . . . . 7 ((𝐵 No 𝐴 No 𝐵 <s 𝐴) → {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)} ∈ (dom 𝐵 ∪ dom 𝐴))
106, 7, 8, 9syl3anc 1364 . . . . . 6 (((𝐴 No 𝐵 No ) ∧ 𝐵 <s 𝐴) → {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)} ∈ (dom 𝐵 ∪ dom 𝐴))
11 necom 3036 . . . . . . . 8 ((𝐴𝑥) ≠ (𝐵𝑥) ↔ (𝐵𝑥) ≠ (𝐴𝑥))
1211rabbii 3418 . . . . . . 7 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} = {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}
1312inteqi 4788 . . . . . 6 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} = {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}
14 uncom 4052 . . . . . 6 (dom 𝐴 ∪ dom 𝐵) = (dom 𝐵 ∪ dom 𝐴)
1510, 13, 143eltr4g 2899 . . . . 5 (((𝐴 No 𝐵 No ) ∧ 𝐵 <s 𝐴) → {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ (dom 𝐴 ∪ dom 𝐵))
165, 15jaodan 952 . . . 4 (((𝐴 No 𝐵 No ) ∧ (𝐴 <s 𝐵𝐵 <s 𝐴)) → {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ (dom 𝐴 ∪ dom 𝐵))
1716ex 413 . . 3 ((𝐴 No 𝐵 No ) → ((𝐴 <s 𝐵𝐵 <s 𝐴) → {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ (dom 𝐴 ∪ dom 𝐵)))
183, 17sylbid 241 . 2 ((𝐴 No 𝐵 No ) → (𝐴𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ (dom 𝐴 ∪ dom 𝐵)))
19183impia 1110 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 207  wa 396  wo 842  w3a 1080  wcel 2080  wne 2983  {crab 3108  cun 3859   cint 4784   class class class wbr 4964   Or wor 5364  dom cdm 5446  Oncon0 6069  cfv 6228   No csur 32750   <s cslt 32751
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343  ax-ext 2768  ax-sep 5097  ax-nul 5104  ax-pow 5160  ax-pr 5224  ax-un 7322
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-mo 2575  df-eu 2611  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ne 2984  df-ral 3109  df-rex 3110  df-rab 3113  df-v 3438  df-sbc 3708  df-csb 3814  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-pss 3878  df-nul 4214  df-if 4384  df-pw 4457  df-sn 4475  df-pr 4477  df-tp 4479  df-op 4481  df-uni 4748  df-int 4785  df-br 4965  df-opab 5027  df-mpt 5044  df-tr 5067  df-id 5351  df-eprel 5356  df-po 5365  df-so 5366  df-fr 5405  df-we 5407  df-xp 5452  df-rel 5453  df-cnv 5454  df-co 5455  df-dm 5456  df-rn 5457  df-res 5458  df-ima 5459  df-ord 6072  df-on 6073  df-suc 6075  df-iota 6192  df-fun 6230  df-fn 6231  df-f 6232  df-fv 6236  df-1o 7956  df-2o 7957  df-no 32753  df-slt 32754
This theorem is referenced by:  nodenselem5  32795  noresle  32803
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