Theorem noextendlt 33289

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 33269	. . . . . . . . 9 ⊢ (𝐴 ∈ No → Fun 𝐴)
2		funfn 6354	. . . . . . . . 9 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
3	1, 2	sylib 221	. . . . . . . 8 ⊢ (𝐴 ∈ No → 𝐴 Fn dom 𝐴)
4		nodmon 33270	. . . . . . . . 9 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On)
5		1on 8092	. . . . . . . . 9 ⊢ 1o ∈ On
6		fnsng 6376	. . . . . . . . 9 ⊢ ((dom 𝐴 ∈ On ∧ 1o ∈ On) → {⟨dom 𝐴, 1o⟩} Fn {dom 𝐴})
7	4, 5, 6	sylancl 589	. . . . . . . 8 ⊢ (𝐴 ∈ No → {⟨dom 𝐴, 1o⟩} Fn {dom 𝐴})
8		nodmord 33273	. . . . . . . . . 10 ⊢ (𝐴 ∈ No → Ord dom 𝐴)
9		ordirr 6177	. . . . . . . . . 10 ⊢ (Ord dom 𝐴 → ¬ dom 𝐴 ∈ dom 𝐴)
10	8, 9	syl 17	. . . . . . . . 9 ⊢ (𝐴 ∈ No → ¬ dom 𝐴 ∈ dom 𝐴)
11		disjsn 4607	. . . . . . . . 9 ⊢ ((dom 𝐴 ∩ {dom 𝐴}) = ∅ ↔ ¬ dom 𝐴 ∈ dom 𝐴)
12	10, 11	sylibr 237	. . . . . . . 8 ⊢ (𝐴 ∈ No → (dom 𝐴 ∩ {dom 𝐴}) = ∅)
13		snidg 4559	. . . . . . . . 9 ⊢ (dom 𝐴 ∈ On → dom 𝐴 ∈ {dom 𝐴})
14	4, 13	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ {dom 𝐴})
15		fvun2 6730	. . . . . . . 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 1371	. . . . . . 7 ⊢ (𝐴 ∈ No → ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘dom 𝐴) = ({⟨dom 𝐴, 1o⟩}‘dom 𝐴))
17		fvsng 6919	. . . . . . . 8 ⊢ ((dom 𝐴 ∈ On ∧ 1o ∈ On) → ({⟨dom 𝐴, 1o⟩}‘dom 𝐴) = 1o)
18	4, 5, 17	sylancl 589	. . . . . . 7 ⊢ (𝐴 ∈ No → ({⟨dom 𝐴, 1o⟩}‘dom 𝐴) = 1o)
19	16, 18	eqtrd 2833	. . . . . 6 ⊢ (𝐴 ∈ No → ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘dom 𝐴) = 1o)
20		ndmfv 6675	. . . . . . 7 ⊢ (¬ dom 𝐴 ∈ dom 𝐴 → (𝐴‘dom 𝐴) = ∅)
21	10, 20	syl 17	. . . . . 6 ⊢ (𝐴 ∈ No → (𝐴‘dom 𝐴) = ∅)
22	19, 21	jca 515	. . . . 5 ⊢ (𝐴 ∈ No → (((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘dom 𝐴) = 1o ∧ (𝐴‘dom 𝐴) = ∅))
23	22	3mix1d 1333	. . . 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 6658	. . . . 5 ⊢ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘dom 𝐴) ∈ V
25		fvex 6658	. . . . 5 ⊢ (𝐴‘dom 𝐴) ∈ V
26	24, 25	brtp 33098	. . . 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 237	. . 3 ⊢ (𝐴 ∈ No → ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘dom 𝐴){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴‘dom 𝐴))
28		necom 3040	. . . . . . 7 ⊢ (((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥) ≠ (𝐴‘𝑥) ↔ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥))
29	28	rabbii 3420	. . . . . 6 ⊢ {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥) ≠ (𝐴‘𝑥)} = {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥)}
30	29	inteqi 4842	. . . . 5 ⊢ ∩ {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥) ≠ (𝐴‘𝑥)} = ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥)}
31		1oex 8093	. . . . . . 7 ⊢ 1o ∈ V
32	31	prid1 4658	. . . . . 6 ⊢ 1o ∈ {1o, 2o}
33	32	noextenddif 33288	. . . . 5 ⊢ (𝐴 ∈ No → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥)} = dom 𝐴)
34	30, 33	syl5eq 2845	. . . 4 ⊢ (𝐴 ∈ No → ∩ {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥) ≠ (𝐴‘𝑥)} = dom 𝐴)
35	34	fveq2d 6649	. . 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 6649	. . 3 ⊢ (𝐴 ∈ No → (𝐴‘∩ {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥) ≠ (𝐴‘𝑥)}) = (𝐴‘dom 𝐴))
37	27, 35, 36	3brtr4d 5062	. 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 33286	. . 3 ⊢ (𝐴 ∈ No → (𝐴 ∪ {⟨dom 𝐴, 1o⟩}) ∈ No )
39		sltval2 33276	. . 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 687	. 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 260	1 ⊢ (𝐴 ∈ No → (𝐴 ∪ {⟨dom 𝐴, 1o⟩}) <s 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ w3o 1083   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  {crab 3110   ∪ cun 3879   ∩ cin 3880  ∅c0 4243  {csn 4525  {ctp 4529  ⟨cop 4531  ∩ cint 4838   class class class wbr 5030  dom cdm 5519  Ord word 6158  Oncon0 6159  Fun wfun 6318   Fn wfn 6319  ‘cfv 6324  1_oc1o 8078  2_oc2o 8079   No csur 33260   <s cslt 33261

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6162  df-on 6163  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-1o 8085  df-2o 8086  df-no 33263  df-slt 33264

This theorem is referenced by: (None)