Theorem noextendgt 31823

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 31806	. . . . . . . 8 ⊢ (𝐴 ∈ No → Ord dom 𝐴)
2		ordirr 5741	. . . . . . . 8 ⊢ (Ord dom 𝐴 → ¬ dom 𝐴 ∈ dom 𝐴)
3	1, 2	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ No → ¬ dom 𝐴 ∈ dom 𝐴)
4		ndmfv 6218	. . . . . . 7 ⊢ (¬ dom 𝐴 ∈ dom 𝐴 → (𝐴‘dom 𝐴) = ∅)
5	3, 4	syl 17	. . . . . 6 ⊢ (𝐴 ∈ No → (𝐴‘dom 𝐴) = ∅)
6		nofun 31802	. . . . . . . . 9 ⊢ (𝐴 ∈ No → Fun 𝐴)
7		funfn 5918	. . . . . . . . 9 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
8	6, 7	sylib 208	. . . . . . . 8 ⊢ (𝐴 ∈ No → 𝐴 Fn dom 𝐴)
9		nodmon 31803	. . . . . . . . 9 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On)
10		2on 7568	. . . . . . . . 9 ⊢ 2𝑜 ∈ On
11		fnsng 5938	. . . . . . . . 9 ⊢ ((dom 𝐴 ∈ On ∧ 2𝑜 ∈ On) → {⟨dom 𝐴, 2𝑜⟩} Fn {dom 𝐴})
12	9, 10, 11	sylancl 694	. . . . . . . 8 ⊢ (𝐴 ∈ No → {⟨dom 𝐴, 2𝑜⟩} Fn {dom 𝐴})
13		disjsn 4246	. . . . . . . . 9 ⊢ ((dom 𝐴 ∩ {dom 𝐴}) = ∅ ↔ ¬ dom 𝐴 ∈ dom 𝐴)
14	3, 13	sylibr 224	. . . . . . . 8 ⊢ (𝐴 ∈ No → (dom 𝐴 ∩ {dom 𝐴}) = ∅)
15		snidg 4206	. . . . . . . . 9 ⊢ (dom 𝐴 ∈ On → dom 𝐴 ∈ {dom 𝐴})
16	9, 15	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ {dom 𝐴})
17		fvun2 6270	. . . . . . . 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 1330	. . . . . . 7 ⊢ (𝐴 ∈ No → ((𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩})‘dom 𝐴) = ({⟨dom 𝐴, 2𝑜⟩}‘dom 𝐴))
19		fvsng 6447	. . . . . . . 8 ⊢ ((dom 𝐴 ∈ On ∧ 2𝑜 ∈ On) → ({⟨dom 𝐴, 2𝑜⟩}‘dom 𝐴) = 2𝑜)
20	9, 10, 19	sylancl 694	. . . . . . 7 ⊢ (𝐴 ∈ No → ({⟨dom 𝐴, 2𝑜⟩}‘dom 𝐴) = 2𝑜)
21	18, 20	eqtrd 2656	. . . . . 6 ⊢ (𝐴 ∈ No → ((𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩})‘dom 𝐴) = 2𝑜)
22	5, 21	jca 554	. . . . 5 ⊢ (𝐴 ∈ No → ((𝐴‘dom 𝐴) = ∅ ∧ ((𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩})‘dom 𝐴) = 2𝑜))
23	22	3mix3d 1238	. . . 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 6201	. . . . 5 ⊢ (𝐴‘dom 𝐴) ∈ V
25		fvex 6201	. . . . 5 ⊢ ((𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩})‘dom 𝐴) ∈ V
26	24, 25	brtp 31639	. . . 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 224	. . 3 ⊢ (𝐴 ∈ No → (𝐴‘dom 𝐴){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} ((𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩})‘dom 𝐴))
28	10	elexi 3213	. . . . . 6 ⊢ 2𝑜 ∈ V
29	28	prid2 4298	. . . . 5 ⊢ 2𝑜 ∈ {1𝑜, 2𝑜}
30	29	noextenddif 31821	. . . 4 ⊢ (𝐴 ∈ No → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩})‘𝑥)} = dom 𝐴)
31	30	fveq2d 6195	. . 3 ⊢ (𝐴 ∈ No → (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩})‘𝑥)}) = (𝐴‘dom 𝐴))
32	30	fveq2d 6195	. . 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 4685	. 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 31819	. . 3 ⊢ (𝐴 ∈ No → (𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩}) ∈ No )
35		sltval2 31809	. . 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 702	. 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 247	1 ⊢ (𝐴 ∈ No → 𝐴 <s (𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∨ w3o 1036   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  {crab 2916   ∪ cun 3572   ∩ cin 3573  ∅c0 3915  {csn 4177  {ctp 4181  ⟨cop 4183  ∩ cint 4475   class class class wbr 4653  dom cdm 5114  Ord word 5722  Oncon0 5723  Fun wfun 5882   Fn wfn 5883  ‘cfv 5888  1_𝑜c1o 7553  2_𝑜c2o 7554   No csur 31793   <s cslt 31794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-1o 7560  df-2o 7561  df-no 31796  df-slt 31797

This theorem is referenced by:  nosupbnd1  31860