Theorem noextendlt 28083

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 28063	. . . . . . . . 9 ⊢ (𝐴 ∈ No → Fun 𝐴)
2		funfn 6598	. . . . . . . . 9 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
3	1, 2	sylib 221	. . . . . . . 8 ⊢ (𝐴 ∈ No → 𝐴 Fn dom 𝐴)
4		nodmon 28064	. . . . . . . . 9 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On)
5		1on 8527	. . . . . . . . 9 ⊢ 1o ∈ On
6		fnsng 6622	. . . . . . . . 9 ⊢ ((dom 𝐴 ∈ On ∧ 1o ∈ On) → {⟨dom 𝐴, 1o⟩} Fn {dom 𝐴})
7	4, 5, 6	sylancl 598	. . . . . . . 8 ⊢ (𝐴 ∈ No → {⟨dom 𝐴, 1o⟩} Fn {dom 𝐴})
8		nodmord 28067	. . . . . . . . . 10 ⊢ (𝐴 ∈ No → Ord dom 𝐴)
9		ordirr 6406	. . . . . . . . . 10 ⊢ (Ord dom 𝐴 → ¬ dom 𝐴 ∈ dom 𝐴)
10	8, 9	syl 18	. . . . . . . . 9 ⊢ (𝐴 ∈ No → ¬ dom 𝐴 ∈ dom 𝐴)
11		disjsn 4673	. . . . . . . . 9 ⊢ ((dom 𝐴 ∩ {dom 𝐴}) = ∅ ↔ ¬ dom 𝐴 ∈ dom 𝐴)
12	10, 11	sylibr 237	. . . . . . . 8 ⊢ (𝐴 ∈ No → (dom 𝐴 ∩ {dom 𝐴}) = ∅)
13		snidg 4622	. . . . . . . . 9 ⊢ (dom 𝐴 ∈ On → dom 𝐴 ∈ {dom 𝐴})
14	4, 13	syl 18	. . . . . . . 8 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ {dom 𝐴})
15		fvun2 7008	. . . . . . . 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 1402	. . . . . . 7 ⊢ (𝐴 ∈ No → ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘dom 𝐴) = ({⟨dom 𝐴, 1o⟩}‘dom 𝐴))
17		fvsng 7216	. . . . . . . 8 ⊢ ((dom 𝐴 ∈ On ∧ 1o ∈ On) → ({⟨dom 𝐴, 1o⟩}‘dom 𝐴) = 1o)
18	4, 5, 17	sylancl 598	. . . . . . 7 ⊢ (𝐴 ∈ No → ({⟨dom 𝐴, 1o⟩}‘dom 𝐴) = 1o)
19	16, 18	eqtrd 2798	. . . . . 6 ⊢ (𝐴 ∈ No → ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘dom 𝐴) = 1o)
20		ndmfv 6948	. . . . . . 7 ⊢ (¬ dom 𝐴 ∈ dom 𝐴 → (𝐴‘dom 𝐴) = ∅)
21	10, 20	syl 18	. . . . . 6 ⊢ (𝐴 ∈ No → (𝐴‘dom 𝐴) = ∅)
22	19, 21	jca 521	. . . . 5 ⊢ (𝐴 ∈ No → (((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘dom 𝐴) = 1o ∧ (𝐴‘dom 𝐴) = ∅))
23	22	3mix1d 1355	. . . 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 6929	. . . . 5 ⊢ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘dom 𝐴) ∈ V
25		fvex 6929	. . . . 5 ⊢ (𝐴‘dom 𝐴) ∈ V
26	24, 25	brtp 5504	. . . 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 3012	. . . . . . 7 ⊢ (((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥) ≠ (𝐴‘𝑥) ↔ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥))
29	28	rabbii 3421	. . . . . 6 ⊢ {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥) ≠ (𝐴‘𝑥)} = {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥)}
30	29	inteqi 4916	. . . . 5 ⊢ ∩ {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥) ≠ (𝐴‘𝑥)} = ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥)}
31		1oex 8524	. . . . . . 7 ⊢ 1o ∈ V
32	31	prid1 4725	. . . . . 6 ⊢ 1o ∈ {1o, 2o}
33	32	noextenddif 28082	. . . . 5 ⊢ (𝐴 ∈ No → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥)} = dom 𝐴)
34	30, 33	eqtrid 2810	. . . 4 ⊢ (𝐴 ∈ No → ∩ {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥) ≠ (𝐴‘𝑥)} = dom 𝐴)
35	34	fveq2d 6920	. . 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 6920	. . 3 ⊢ (𝐴 ∈ No → (𝐴‘∩ {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥) ≠ (𝐴‘𝑥)}) = (𝐴‘dom 𝐴))
37	27, 35, 36	3brtr4d 5143	. 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 28080	. . 3 ⊢ (𝐴 ∈ No → (𝐴 ∪ {⟨dom 𝐴, 1o⟩}) ∈ No)
39		ltsval2 28070	. . 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 701	. 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)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ w3o 1102   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  {crab 3416   ∩ cin 4092   ∪ cun 4152  ∅c0 4281  {csn 4588  {ctp 4592  ⟨cop 4840  ∩ cint 4912   class class class wbr 5108  dom cdm 5658  Ord word 6387  Oncon0 6388  Fun wfun 6560   Fn wfn 6561  ‘cfv 6566  1_oc1o 8507  2_oc2o 8508  Nocsur 28054   <s clts 28055

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-uni 4873  df-int 4913  df-br 5109  df-opab 5174  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-ord 6391  df-on 6392  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fv 6574  df-1o 8514  df-2o 8515  df-no 28057  df-lts 28058

This theorem is used by:  noinfbnd1  28143