Theorem noextendgt 27615

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 27598	. . . . . . . 8 ⊢ (𝐴 ∈ No → Ord dom 𝐴)
2		ordirr 6330	. . . . . . . 8 ⊢ (Ord dom 𝐴 → ¬ dom 𝐴 ∈ dom 𝐴)
3	1, 2	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ No → ¬ dom 𝐴 ∈ dom 𝐴)
4		ndmfv 6860	. . . . . . 7 ⊢ (¬ dom 𝐴 ∈ dom 𝐴 → (𝐴‘dom 𝐴) = ∅)
5	3, 4	syl 17	. . . . . 6 ⊢ (𝐴 ∈ No → (𝐴‘dom 𝐴) = ∅)
6		nofun 27594	. . . . . . . . 9 ⊢ (𝐴 ∈ No → Fun 𝐴)
7		funfn 6517	. . . . . . . . 9 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
8	6, 7	sylib 218	. . . . . . . 8 ⊢ (𝐴 ∈ No → 𝐴 Fn dom 𝐴)
9		nodmon 27595	. . . . . . . . 9 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On)
10		2on 8404	. . . . . . . . 9 ⊢ 2o ∈ On
11		fnsng 6539	. . . . . . . . 9 ⊢ ((dom 𝐴 ∈ On ∧ 2o ∈ On) → {⟨dom 𝐴, 2o⟩} Fn {dom 𝐴})
12	9, 10, 11	sylancl 586	. . . . . . . 8 ⊢ (𝐴 ∈ No → {⟨dom 𝐴, 2o⟩} Fn {dom 𝐴})
13		disjsn 4663	. . . . . . . . 9 ⊢ ((dom 𝐴 ∩ {dom 𝐴}) = ∅ ↔ ¬ dom 𝐴 ∈ dom 𝐴)
14	3, 13	sylibr 234	. . . . . . . 8 ⊢ (𝐴 ∈ No → (dom 𝐴 ∩ {dom 𝐴}) = ∅)
15		snidg 4612	. . . . . . . . 9 ⊢ (dom 𝐴 ∈ On → dom 𝐴 ∈ {dom 𝐴})
16	9, 15	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ {dom 𝐴})
17		fvun2 6920	. . . . . . . 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 1376	. . . . . . 7 ⊢ (𝐴 ∈ No → ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘dom 𝐴) = ({⟨dom 𝐴, 2o⟩}‘dom 𝐴))
19		fvsng 7120	. . . . . . . 8 ⊢ ((dom 𝐴 ∈ On ∧ 2o ∈ On) → ({⟨dom 𝐴, 2o⟩}‘dom 𝐴) = 2o)
20	9, 10, 19	sylancl 586	. . . . . . 7 ⊢ (𝐴 ∈ No → ({⟨dom 𝐴, 2o⟩}‘dom 𝐴) = 2o)
21	18, 20	eqtrd 2766	. . . . . 6 ⊢ (𝐴 ∈ No → ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘dom 𝐴) = 2o)
22	5, 21	jca 511	. . . . 5 ⊢ (𝐴 ∈ No → ((𝐴‘dom 𝐴) = ∅ ∧ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘dom 𝐴) = 2o))
23	22	3mix3d 1339	. . . 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 6841	. . . . 5 ⊢ (𝐴‘dom 𝐴) ∈ V
25		fvex 6841	. . . . 5 ⊢ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘dom 𝐴) ∈ V
26	24, 25	brtp 5466	. . . 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 234	. . 3 ⊢ (𝐴 ∈ No → (𝐴‘dom 𝐴){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘dom 𝐴))
28	10	elexi 3459	. . . . . 6 ⊢ 2o ∈ V
29	28	prid2 4715	. . . . 5 ⊢ 2o ∈ {1o, 2o}
30	29	noextenddif 27613	. . . 4 ⊢ (𝐴 ∈ No → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘𝑥)} = dom 𝐴)
31	30	fveq2d 6832	. . 3 ⊢ (𝐴 ∈ No → (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘𝑥)}) = (𝐴‘dom 𝐴))
32	30	fveq2d 6832	. . 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 5125	. 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 27611	. . 3 ⊢ (𝐴 ∈ No → (𝐴 ∪ {⟨dom 𝐴, 2o⟩}) ∈ No )
35		sltval2 27601	. . 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 687	. 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 257	1 ⊢ (𝐴 ∈ No → 𝐴 <s (𝐴 ∪ {⟨dom 𝐴, 2o⟩}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ w3o 1085   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  {crab 3395   ∪ cun 3895   ∩ cin 3896  ∅c0 4282  {csn 4575  {ctp 4579  ⟨cop 4581  ∩ cint 4897   class class class wbr 5093  dom cdm 5619  Ord word 6311  Oncon0 6312  Fun wfun 6481   Fn wfn 6482  ‘cfv 6487  1_oc1o 8384  2_oc2o 8385   No csur 27584   <s cslt 27585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-tp 4580  df-op 4582  df-uni 4859  df-int 4898  df-br 5094  df-opab 5156  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6315  df-on 6316  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-fv 6495  df-1o 8391  df-2o 8392  df-no 27587  df-slt 27588

This theorem is referenced by:  nosupbnd1  27659