Theorem noextendlt 31947

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 31927	. . . . . . . . 9 ⊢ (𝐴 ∈ No → Fun 𝐴)
2		funfn 5956	. . . . . . . . 9 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
3	1, 2	sylib 208	. . . . . . . 8 ⊢ (𝐴 ∈ No → 𝐴 Fn dom 𝐴)
4		nodmon 31928	. . . . . . . . 9 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On)
5		1on 7612	. . . . . . . . 9 ⊢ 1𝑜 ∈ On
6		fnsng 5976	. . . . . . . . 9 ⊢ ((dom 𝐴 ∈ On ∧ 1𝑜 ∈ On) → {⟨dom 𝐴, 1𝑜⟩} Fn {dom 𝐴})
7	4, 5, 6	sylancl 695	. . . . . . . 8 ⊢ (𝐴 ∈ No → {⟨dom 𝐴, 1𝑜⟩} Fn {dom 𝐴})
8		nodmord 31931	. . . . . . . . . 10 ⊢ (𝐴 ∈ No → Ord dom 𝐴)
9		ordirr 5779	. . . . . . . . . 10 ⊢ (Ord dom 𝐴 → ¬ dom 𝐴 ∈ dom 𝐴)
10	8, 9	syl 17	. . . . . . . . 9 ⊢ (𝐴 ∈ No → ¬ dom 𝐴 ∈ dom 𝐴)
11		disjsn 4278	. . . . . . . . 9 ⊢ ((dom 𝐴 ∩ {dom 𝐴}) = ∅ ↔ ¬ dom 𝐴 ∈ dom 𝐴)
12	10, 11	sylibr 224	. . . . . . . 8 ⊢ (𝐴 ∈ No → (dom 𝐴 ∩ {dom 𝐴}) = ∅)
13		snidg 4239	. . . . . . . . 9 ⊢ (dom 𝐴 ∈ On → dom 𝐴 ∈ {dom 𝐴})
14	4, 13	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ {dom 𝐴})
15		fvun2 6309	. . . . . . . 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 1370	. . . . . . 7 ⊢ (𝐴 ∈ No → ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴) = ({⟨dom 𝐴, 1𝑜⟩}‘dom 𝐴))
17		fvsng 6488	. . . . . . . 8 ⊢ ((dom 𝐴 ∈ On ∧ 1𝑜 ∈ On) → ({⟨dom 𝐴, 1𝑜⟩}‘dom 𝐴) = 1𝑜)
18	4, 5, 17	sylancl 695	. . . . . . 7 ⊢ (𝐴 ∈ No → ({⟨dom 𝐴, 1𝑜⟩}‘dom 𝐴) = 1𝑜)
19	16, 18	eqtrd 2685	. . . . . 6 ⊢ (𝐴 ∈ No → ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴) = 1𝑜)
20		ndmfv 6256	. . . . . . 7 ⊢ (¬ dom 𝐴 ∈ dom 𝐴 → (𝐴‘dom 𝐴) = ∅)
21	10, 20	syl 17	. . . . . 6 ⊢ (𝐴 ∈ No → (𝐴‘dom 𝐴) = ∅)
22	19, 21	jca 553	. . . . 5 ⊢ (𝐴 ∈ No → (((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴) = 1𝑜 ∧ (𝐴‘dom 𝐴) = ∅))
23	22	3mix1d 1256	. . . 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 6239	. . . . 5 ⊢ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴) ∈ V
25		fvex 6239	. . . . 5 ⊢ (𝐴‘dom 𝐴) ∈ V
26	24, 25	brtp 31765	. . . 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 2876	. . . . . . 7 ⊢ (((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥) ≠ (𝐴‘𝑥) ↔ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥))
29	28	rabbii 3216	. . . . . 6 ⊢ {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥) ≠ (𝐴‘𝑥)} = {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥)}
30	29	inteqi 4511	. . . . 5 ⊢ ∩ {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥) ≠ (𝐴‘𝑥)} = ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥)}
31	5	elexi 3244	. . . . . . 7 ⊢ 1𝑜 ∈ V
32	31	prid1 4329	. . . . . 6 ⊢ 1𝑜 ∈ {1𝑜, 2𝑜}
33	32	noextenddif 31946	. . . . 5 ⊢ (𝐴 ∈ No → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥)} = dom 𝐴)
34	30, 33	syl5eq 2697	. . . 4 ⊢ (𝐴 ∈ No → ∩ {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥) ≠ (𝐴‘𝑥)} = dom 𝐴)
35	34	fveq2d 6233	. . 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 6233	. . 3 ⊢ (𝐴 ∈ No → (𝐴‘∩ {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥) ≠ (𝐴‘𝑥)}) = (𝐴‘dom 𝐴))
37	27, 35, 36	3brtr4d 4717	. 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 31944	. . 3 ⊢ (𝐴 ∈ No → (𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩}) ∈ No )
39		sltval2 31934	. . 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 704	. 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 247	1 ⊢ (𝐴 ∈ No → (𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩}) <s 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   ∨ w3o 1053   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  {crab 2945   ∪ cun 3605   ∩ cin 3606  ∅c0 3948  {csn 4210  {ctp 4214  ⟨cop 4216  ∩ cint 4507   class class class wbr 4685  dom cdm 5143  Ord word 5760  Oncon0 5761  Fun wfun 5920   Fn wfn 5921  ‘cfv 5926  1_𝑜c1o 7598  2_𝑜c2o 7599   No csur 31918   <s cslt 31919

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-1o 7605  df-2o 7606  df-no 31921  df-slt 31922

This theorem is referenced by: (None)