Theorem noextendlt 33872

Description: Extending a surreal with a negative sign results in a smaller surreal. (Contributed by Scott Fenton, 22-Nov-2021.)

Assertion

Ref	Expression
noextendlt	⊢ (𝐴 ∈ No → (𝐴 ∪ {⟨dom 𝐴, 1o⟩}) <s 𝐴)

Proof of Theorem noextendlt

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nofun 33852	. . . . . . . . 9 ⊢ (𝐴 ∈ No → Fun 𝐴)
2		funfn 6464	. . . . . . . . 9 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
3	1, 2	sylib 217	. . . . . . . 8 ⊢ (𝐴 ∈ No → 𝐴 Fn dom 𝐴)
4		nodmon 33853	. . . . . . . . 9 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On)
5		1on 8309	. . . . . . . . 9 ⊢ 1o ∈ On
6		fnsng 6486	. . . . . . . . 9 ⊢ ((dom 𝐴 ∈ On ∧ 1o ∈ On) → {⟨dom 𝐴, 1o⟩} Fn {dom 𝐴})
7	4, 5, 6	sylancl 586	. . . . . . . 8 ⊢ (𝐴 ∈ No → {⟨dom 𝐴, 1o⟩} Fn {dom 𝐴})
8		nodmord 33856	. . . . . . . . . 10 ⊢ (𝐴 ∈ No → Ord dom 𝐴)
9		ordirr 6284	. . . . . . . . . 10 ⊢ (Ord dom 𝐴 → ¬ dom 𝐴 ∈ dom 𝐴)
10	8, 9	syl 17	. . . . . . . . 9 ⊢ (𝐴 ∈ No → ¬ dom 𝐴 ∈ dom 𝐴)
11		disjsn 4647	. . . . . . . . 9 ⊢ ((dom 𝐴 ∩ {dom 𝐴}) = ∅ ↔ ¬ dom 𝐴 ∈ dom 𝐴)
12	10, 11	sylibr 233	. . . . . . . 8 ⊢ (𝐴 ∈ No → (dom 𝐴 ∩ {dom 𝐴}) = ∅)
13		snidg 4595	. . . . . . . . 9 ⊢ (dom 𝐴 ∈ On → dom 𝐴 ∈ {dom 𝐴})
14	4, 13	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ {dom 𝐴})
15		fvun2 6860	. . . . . . . 8 ⊢ ((𝐴 Fn dom 𝐴 ∧ {⟨dom 𝐴, 1o⟩} Fn {dom 𝐴} ∧ ((dom 𝐴 ∩ {dom 𝐴}) = ∅ ∧ dom 𝐴 ∈ {dom 𝐴})) → ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘dom 𝐴) = ({⟨dom 𝐴, 1o⟩}‘dom 𝐴))
16	3, 7, 12, 14, 15	syl112anc 1373	. . . . . . 7 ⊢ (𝐴 ∈ No → ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘dom 𝐴) = ({⟨dom 𝐴, 1o⟩}‘dom 𝐴))
17		fvsng 7052	. . . . . . . 8 ⊢ ((dom 𝐴 ∈ On ∧ 1o ∈ On) → ({⟨dom 𝐴, 1o⟩}‘dom 𝐴) = 1o)
18	4, 5, 17	sylancl 586	. . . . . . 7 ⊢ (𝐴 ∈ No → ({⟨dom 𝐴, 1o⟩}‘dom 𝐴) = 1o)
19	16, 18	eqtrd 2778	. . . . . 6 ⊢ (𝐴 ∈ No → ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘dom 𝐴) = 1o)
20		ndmfv 6804	. . . . . . 7 ⊢ (¬ dom 𝐴 ∈ dom 𝐴 → (𝐴‘dom 𝐴) = ∅)
21	10, 20	syl 17	. . . . . 6 ⊢ (𝐴 ∈ No → (𝐴‘dom 𝐴) = ∅)
22	19, 21	jca 512	. . . . 5 ⊢ (𝐴 ∈ No → (((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘dom 𝐴) = 1o ∧ (𝐴‘dom 𝐴) = ∅))
23	22	3mix1d 1335	. . . 4 ⊢ (𝐴 ∈ No → ((((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘dom 𝐴) = 1o ∧ (𝐴‘dom 𝐴) = ∅) ∨ (((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘dom 𝐴) = 1o ∧ (𝐴‘dom 𝐴) = 2o) ∨ (((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘dom 𝐴) = ∅ ∧ (𝐴‘dom 𝐴) = 2o)))
24		fvex 6787	. . . . 5 ⊢ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘dom 𝐴) ∈ V
25		fvex 6787	. . . . 5 ⊢ (𝐴‘dom 𝐴) ∈ V
26	24, 25	brtp 33717	. . . 4 ⊢ (((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘dom 𝐴){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴‘dom 𝐴) ↔ ((((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘dom 𝐴) = 1o ∧ (𝐴‘dom 𝐴) = ∅) ∨ (((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘dom 𝐴) = 1o ∧ (𝐴‘dom 𝐴) = 2o) ∨ (((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘dom 𝐴) = ∅ ∧ (𝐴‘dom 𝐴) = 2o)))
27	23, 26	sylibr 233	. . 3 ⊢ (𝐴 ∈ No → ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘dom 𝐴){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴‘dom 𝐴))
28		necom 2997	. . . . . . 7 ⊢ (((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥) ≠ (𝐴‘𝑥) ↔ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥))
29	28	rabbii 3408	. . . . . 6 ⊢ {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥) ≠ (𝐴‘𝑥)} = {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥)}
30	29	inteqi 4883	. . . . 5 ⊢ ∩ {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥) ≠ (𝐴‘𝑥)} = ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥)}
31		1oex 8307	. . . . . . 7 ⊢ 1o ∈ V
32	31	prid1 4698	. . . . . 6 ⊢ 1o ∈ {1o, 2o}
33	32	noextenddif 33871	. . . . 5 ⊢ (𝐴 ∈ No → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥)} = dom 𝐴)
34	30, 33	eqtrid 2790	. . . 4 ⊢ (𝐴 ∈ No → ∩ {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥) ≠ (𝐴‘𝑥)} = dom 𝐴)
35	34	fveq2d 6778	. . 3 ⊢ (𝐴 ∈ No → ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘∩ {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥) ≠ (𝐴‘𝑥)}) = ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘dom 𝐴))
36	34	fveq2d 6778	. . 3 ⊢ (𝐴 ∈ No → (𝐴‘∩ {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥) ≠ (𝐴‘𝑥)}) = (𝐴‘dom 𝐴))
37	27, 35, 36	3brtr4d 5106	. 2 ⊢ (𝐴 ∈ No → ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘∩ {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥) ≠ (𝐴‘𝑥)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴‘∩ {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥) ≠ (𝐴‘𝑥)}))
38	32	noextend 33869	. . 3 ⊢ (𝐴 ∈ No → (𝐴 ∪ {⟨dom 𝐴, 1o⟩}) ∈ No )
39		sltval2 33859	. . 3 ⊢ (((𝐴 ∪ {⟨dom 𝐴, 1o⟩}) ∈ No ∧ 𝐴 ∈ No ) → ((𝐴 ∪ {⟨dom 𝐴, 1o⟩}) <s 𝐴 ↔ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘∩ {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥) ≠ (𝐴‘𝑥)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴‘∩ {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥) ≠ (𝐴‘𝑥)})))
40	38, 39	mpancom 685	. 2 ⊢ (𝐴 ∈ No → ((𝐴 ∪ {⟨dom 𝐴, 1o⟩}) <s 𝐴 ↔ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘∩ {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥) ≠ (𝐴‘𝑥)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴‘∩ {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥) ≠ (𝐴‘𝑥)})))
41	37, 40	mpbird 256	1 ⊢ (𝐴 ∈ No → (𝐴 ∪ {⟨dom 𝐴, 1o⟩}) <s 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ w3o 1085   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  {crab 3068   ∪ cun 3885   ∩ cin 3886  ∅c0 4256  {csn 4561  {ctp 4565  ⟨cop 4567  ∩ cint 4879   class class class wbr 5074  dom cdm 5589  Ord word 6265  Oncon0 6266  Fun wfun 6427   Fn wfn 6428  ‘cfv 6433  1_oc1o 8290  2_oc2o 8291   No csur 33843   <s cslt 33844

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-1o 8297  df-2o 8298  df-no 33846  df-slt 33847

This theorem is referenced by:  noinfbnd1  33932