Theorem noextendlt 31822

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 31802	. . . . . . . . 9 ⊢ (𝐴 ∈ No → Fun 𝐴)
2		funfn 5918	. . . . . . . . 9 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
3	1, 2	sylib 208	. . . . . . . 8 ⊢ (𝐴 ∈ No → 𝐴 Fn dom 𝐴)
4		nodmon 31803	. . . . . . . . 9 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On)
5		1on 7567	. . . . . . . . 9 ⊢ 1𝑜 ∈ On
6		fnsng 5938	. . . . . . . . 9 ⊢ ((dom 𝐴 ∈ On ∧ 1𝑜 ∈ On) → {⟨dom 𝐴, 1𝑜⟩} Fn {dom 𝐴})
7	4, 5, 6	sylancl 694	. . . . . . . 8 ⊢ (𝐴 ∈ No → {⟨dom 𝐴, 1𝑜⟩} Fn {dom 𝐴})
8		nodmord 31806	. . . . . . . . . 10 ⊢ (𝐴 ∈ No → Ord dom 𝐴)
9		ordirr 5741	. . . . . . . . . 10 ⊢ (Ord dom 𝐴 → ¬ dom 𝐴 ∈ dom 𝐴)
10	8, 9	syl 17	. . . . . . . . 9 ⊢ (𝐴 ∈ No → ¬ dom 𝐴 ∈ dom 𝐴)
11		disjsn 4246	. . . . . . . . 9 ⊢ ((dom 𝐴 ∩ {dom 𝐴}) = ∅ ↔ ¬ dom 𝐴 ∈ dom 𝐴)
12	10, 11	sylibr 224	. . . . . . . 8 ⊢ (𝐴 ∈ No → (dom 𝐴 ∩ {dom 𝐴}) = ∅)
13		snidg 4206	. . . . . . . . 9 ⊢ (dom 𝐴 ∈ On → dom 𝐴 ∈ {dom 𝐴})
14	4, 13	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ {dom 𝐴})
15		fvun2 6270	. . . . . . . 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 1330	. . . . . . 7 ⊢ (𝐴 ∈ No → ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴) = ({⟨dom 𝐴, 1𝑜⟩}‘dom 𝐴))
17		fvsng 6447	. . . . . . . 8 ⊢ ((dom 𝐴 ∈ On ∧ 1𝑜 ∈ On) → ({⟨dom 𝐴, 1𝑜⟩}‘dom 𝐴) = 1𝑜)
18	4, 5, 17	sylancl 694	. . . . . . 7 ⊢ (𝐴 ∈ No → ({⟨dom 𝐴, 1𝑜⟩}‘dom 𝐴) = 1𝑜)
19	16, 18	eqtrd 2656	. . . . . 6 ⊢ (𝐴 ∈ No → ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴) = 1𝑜)
20		ndmfv 6218	. . . . . . 7 ⊢ (¬ dom 𝐴 ∈ dom 𝐴 → (𝐴‘dom 𝐴) = ∅)
21	10, 20	syl 17	. . . . . 6 ⊢ (𝐴 ∈ No → (𝐴‘dom 𝐴) = ∅)
22	19, 21	jca 554	. . . . 5 ⊢ (𝐴 ∈ No → (((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴) = 1𝑜 ∧ (𝐴‘dom 𝐴) = ∅))
23	22	3mix1d 1236	. . . 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 6201	. . . . 5 ⊢ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴) ∈ V
25		fvex 6201	. . . . 5 ⊢ (𝐴‘dom 𝐴) ∈ V
26	24, 25	brtp 31639	. . . 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 224	. . 3 ⊢ (𝐴 ∈ No → ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴‘dom 𝐴))
28		necom 2847	. . . . . . . 8 ⊢ (((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥) ≠ (𝐴‘𝑥) ↔ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥))
29	28	a1i 11	. . . . . . 7 ⊢ (𝑥 ∈ On → (((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥) ≠ (𝐴‘𝑥) ↔ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥)))
30	29	rabbiia 3185	. . . . . 6 ⊢ {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥) ≠ (𝐴‘𝑥)} = {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥)}
31	30	inteqi 4479	. . . . 5 ⊢ ∩ {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥) ≠ (𝐴‘𝑥)} = ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥)}
32	5	elexi 3213	. . . . . . 7 ⊢ 1𝑜 ∈ V
33	32	prid1 4297	. . . . . 6 ⊢ 1𝑜 ∈ {1𝑜, 2𝑜}
34	33	noextenddif 31821	. . . . 5 ⊢ (𝐴 ∈ No → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥)} = dom 𝐴)
35	31, 34	syl5eq 2668	. . . 4 ⊢ (𝐴 ∈ No → ∩ {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥) ≠ (𝐴‘𝑥)} = dom 𝐴)
36	35	fveq2d 6195	. . 3 ⊢ (𝐴 ∈ No → ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘∩ {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥) ≠ (𝐴‘𝑥)}) = ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴))
37	35	fveq2d 6195	. . 3 ⊢ (𝐴 ∈ No → (𝐴‘∩ {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥) ≠ (𝐴‘𝑥)}) = (𝐴‘dom 𝐴))
38	27, 36, 37	3brtr4d 4685	. 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)}))
39	33	noextend 31819	. . 3 ⊢ (𝐴 ∈ No → (𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩}) ∈ No )
40		sltval2 31809	. . 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)})))
41	39, 40	mpancom 703	. 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)})))
42	38, 41	mpbird 247	1 ⊢ (𝐴 ∈ No → (𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩}) <s 𝐴)

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: (None)