Theorem noextendgt 27639

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

Assertion

Ref	Expression
noextendgt	⊢ (𝐴 ∈ No → 𝐴 <s (𝐴 ∪ {⟨dom 𝐴, 2o⟩}))

Proof of Theorem noextendgt

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nodmord 27622	. . . . . . . 8 ⊢ (𝐴 ∈ No → Ord dom 𝐴)
2		ordirr 6375	. . . . . . . 8 ⊢ (Ord dom 𝐴 → ¬ dom 𝐴 ∈ dom 𝐴)
3	1, 2	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ No → ¬ dom 𝐴 ∈ dom 𝐴)
4		ndmfv 6916	. . . . . . 7 ⊢ (¬ dom 𝐴 ∈ dom 𝐴 → (𝐴‘dom 𝐴) = ∅)
5	3, 4	syl 17	. . . . . 6 ⊢ (𝐴 ∈ No → (𝐴‘dom 𝐴) = ∅)
6		nofun 27618	. . . . . . . . 9 ⊢ (𝐴 ∈ No → Fun 𝐴)
7		funfn 6571	. . . . . . . . 9 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
8	6, 7	sylib 218	. . . . . . . 8 ⊢ (𝐴 ∈ No → 𝐴 Fn dom 𝐴)
9		nodmon 27619	. . . . . . . . 9 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On)
10		2on 8499	. . . . . . . . 9 ⊢ 2o ∈ On
11		fnsng 6593	. . . . . . . . 9 ⊢ ((dom 𝐴 ∈ On ∧ 2o ∈ On) → {⟨dom 𝐴, 2o⟩} Fn {dom 𝐴})
12	9, 10, 11	sylancl 586	. . . . . . . 8 ⊢ (𝐴 ∈ No → {⟨dom 𝐴, 2o⟩} Fn {dom 𝐴})
13		disjsn 4692	. . . . . . . . 9 ⊢ ((dom 𝐴 ∩ {dom 𝐴}) = ∅ ↔ ¬ dom 𝐴 ∈ dom 𝐴)
14	3, 13	sylibr 234	. . . . . . . 8 ⊢ (𝐴 ∈ No → (dom 𝐴 ∩ {dom 𝐴}) = ∅)
15		snidg 4641	. . . . . . . . 9 ⊢ (dom 𝐴 ∈ On → dom 𝐴 ∈ {dom 𝐴})
16	9, 15	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ {dom 𝐴})
17		fvun2 6976	. . . . . . . 8 ⊢ ((𝐴 Fn dom 𝐴 ∧ {⟨dom 𝐴, 2o⟩} Fn {dom 𝐴} ∧ ((dom 𝐴 ∩ {dom 𝐴}) = ∅ ∧ dom 𝐴 ∈ {dom 𝐴})) → ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘dom 𝐴) = ({⟨dom 𝐴, 2o⟩}‘dom 𝐴))
18	8, 12, 14, 16, 17	syl112anc 1376	. . . . . . 7 ⊢ (𝐴 ∈ No → ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘dom 𝐴) = ({⟨dom 𝐴, 2o⟩}‘dom 𝐴))
19		fvsng 7177	. . . . . . . 8 ⊢ ((dom 𝐴 ∈ On ∧ 2o ∈ On) → ({⟨dom 𝐴, 2o⟩}‘dom 𝐴) = 2o)
20	9, 10, 19	sylancl 586	. . . . . . 7 ⊢ (𝐴 ∈ No → ({⟨dom 𝐴, 2o⟩}‘dom 𝐴) = 2o)
21	18, 20	eqtrd 2771	. . . . . 6 ⊢ (𝐴 ∈ No → ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘dom 𝐴) = 2o)
22	5, 21	jca 511	. . . . 5 ⊢ (𝐴 ∈ No → ((𝐴‘dom 𝐴) = ∅ ∧ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘dom 𝐴) = 2o))
23	22	3mix3d 1339	. . . 4 ⊢ (𝐴 ∈ No → (((𝐴‘dom 𝐴) = 1o ∧ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘dom 𝐴) = ∅) ∨ ((𝐴‘dom 𝐴) = 1o ∧ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘dom 𝐴) = 2o) ∨ ((𝐴‘dom 𝐴) = ∅ ∧ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘dom 𝐴) = 2o)))
24		fvex 6894	. . . . 5 ⊢ (𝐴‘dom 𝐴) ∈ V
25		fvex 6894	. . . . 5 ⊢ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘dom 𝐴) ∈ V
26	24, 25	brtp 5503	. . . 4 ⊢ ((𝐴‘dom 𝐴){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘dom 𝐴) ↔ (((𝐴‘dom 𝐴) = 1o ∧ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘dom 𝐴) = ∅) ∨ ((𝐴‘dom 𝐴) = 1o ∧ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘dom 𝐴) = 2o) ∨ ((𝐴‘dom 𝐴) = ∅ ∧ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘dom 𝐴) = 2o)))
27	23, 26	sylibr 234	. . 3 ⊢ (𝐴 ∈ No → (𝐴‘dom 𝐴){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘dom 𝐴))
28	10	elexi 3487	. . . . . 6 ⊢ 2o ∈ V
29	28	prid2 4744	. . . . 5 ⊢ 2o ∈ {1o, 2o}
30	29	noextenddif 27637	. . . 4 ⊢ (𝐴 ∈ No → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘𝑥)} = dom 𝐴)
31	30	fveq2d 6885	. . 3 ⊢ (𝐴 ∈ No → (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘𝑥)}) = (𝐴‘dom 𝐴))
32	30	fveq2d 6885	. . 3 ⊢ (𝐴 ∈ No → ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘𝑥)}) = ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘dom 𝐴))
33	27, 31, 32	3brtr4d 5156	. 2 ⊢ (𝐴 ∈ No → (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘𝑥)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘𝑥)}))
34	29	noextend 27635	. . 3 ⊢ (𝐴 ∈ No → (𝐴 ∪ {⟨dom 𝐴, 2o⟩}) ∈ No )
35		sltval2 27625	. . 3 ⊢ ((𝐴 ∈ No ∧ (𝐴 ∪ {⟨dom 𝐴, 2o⟩}) ∈ No ) → (𝐴 <s (𝐴 ∪ {⟨dom 𝐴, 2o⟩}) ↔ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘𝑥)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘𝑥)})))
36	34, 35	mpdan 687	. 2 ⊢ (𝐴 ∈ No → (𝐴 <s (𝐴 ∪ {⟨dom 𝐴, 2o⟩}) ↔ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘𝑥)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘𝑥)})))
37	33, 36	mpbird 257	1 ⊢ (𝐴 ∈ No → 𝐴 <s (𝐴 ∪ {⟨dom 𝐴, 2o⟩}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ w3o 1085   = wceq 1540   ∈ wcel 2109   ≠ wne 2933  {crab 3420   ∪ cun 3929   ∩ cin 3930  ∅c0 4313  {csn 4606  {ctp 4610  ⟨cop 4612  ∩ cint 4927   class class class wbr 5124  dom cdm 5659  Ord word 6356  Oncon0 6357  Fun wfun 6530   Fn wfn 6531  ‘cfv 6536  1_oc1o 8478  2_oc2o 8479   No csur 27608   <s cslt 27609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-uni 4889  df-int 4928  df-br 5125  df-opab 5187  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6360  df-on 6361  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-1o 8485  df-2o 8486  df-no 27611  df-slt 27612

This theorem is referenced by:  nosupbnd1  27683