Theorem noextendgt 27730

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 27713	. . . . . . . 8 ⊢ (𝐴 ∈ No → Ord dom 𝐴)
2		ordirr 6404	. . . . . . . 8 ⊢ (Ord dom 𝐴 → ¬ dom 𝐴 ∈ dom 𝐴)
3	1, 2	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ No → ¬ dom 𝐴 ∈ dom 𝐴)
4		ndmfv 6942	. . . . . . 7 ⊢ (¬ dom 𝐴 ∈ dom 𝐴 → (𝐴‘dom 𝐴) = ∅)
5	3, 4	syl 17	. . . . . 6 ⊢ (𝐴 ∈ No → (𝐴‘dom 𝐴) = ∅)
6		nofun 27709	. . . . . . . . 9 ⊢ (𝐴 ∈ No → Fun 𝐴)
7		funfn 6598	. . . . . . . . 9 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
8	6, 7	sylib 218	. . . . . . . 8 ⊢ (𝐴 ∈ No → 𝐴 Fn dom 𝐴)
9		nodmon 27710	. . . . . . . . 9 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On)
10		2on 8519	. . . . . . . . 9 ⊢ 2o ∈ On
11		fnsng 6620	. . . . . . . . 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 4716	. . . . . . . . 9 ⊢ ((dom 𝐴 ∩ {dom 𝐴}) = ∅ ↔ ¬ dom 𝐴 ∈ dom 𝐴)
14	3, 13	sylibr 234	. . . . . . . 8 ⊢ (𝐴 ∈ No → (dom 𝐴 ∩ {dom 𝐴}) = ∅)
15		snidg 4665	. . . . . . . . 9 ⊢ (dom 𝐴 ∈ On → dom 𝐴 ∈ {dom 𝐴})
16	9, 15	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ {dom 𝐴})
17		fvun2 7001	. . . . . . . 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 1373	. . . . . . 7 ⊢ (𝐴 ∈ No → ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘dom 𝐴) = ({⟨dom 𝐴, 2o⟩}‘dom 𝐴))
19		fvsng 7200	. . . . . . . 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 2775	. . . . . 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 1337	. . . 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 6920	. . . . 5 ⊢ (𝐴‘dom 𝐴) ∈ V
25		fvex 6920	. . . . 5 ⊢ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘dom 𝐴) ∈ V
26	24, 25	brtp 5533	. . . 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 3501	. . . . . 6 ⊢ 2o ∈ V
29	28	prid2 4768	. . . . 5 ⊢ 2o ∈ {1o, 2o}
30	29	noextenddif 27728	. . . 4 ⊢ (𝐴 ∈ No → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘𝑥)} = dom 𝐴)
31	30	fveq2d 6911	. . 3 ⊢ (𝐴 ∈ No → (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘𝑥)}) = (𝐴‘dom 𝐴))
32	30	fveq2d 6911	. . 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 5180	. 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 27726	. . 3 ⊢ (𝐴 ∈ No → (𝐴 ∪ {⟨dom 𝐴, 2o⟩}) ∈ No )
35		sltval2 27716	. . 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 1537   ∈ wcel 2106   ≠ wne 2938  {crab 3433   ∪ cun 3961   ∩ cin 3962  ∅c0 4339  {csn 4631  {ctp 4635  ⟨cop 4637  ∩ cint 4951   class class class wbr 5148  dom cdm 5689  Ord word 6385  Oncon0 6386  Fun wfun 6557   Fn wfn 6558  ‘cfv 6563  1_oc1o 8498  2_oc2o 8499   No csur 27699   <s cslt 27700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-1o 8505  df-2o 8506  df-no 27702  df-slt 27703

This theorem is referenced by:  nosupbnd1  27774