Theorem noextend 28029

Description: Extending a surreal by one sign value results in a new surreal. (Contributed by Scott Fenton, 22-Nov-2021.)

Hypothesis

Ref	Expression
noextend.1	⊢ 𝑋 ∈ {1o, 2o}

Assertion

Ref	Expression
noextend	⊢ (𝐴 ∈ No → (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) ∈ No)

Proof of Theorem noextend

Step	Hyp	Ref	Expression
1		nofun 28012	. . 3 ⊢ (𝐴 ∈ No → Fun 𝐴)
2		dmexg 7914	. . . 4 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ V)
3		noextend.1	. . . 4 ⊢ 𝑋 ∈ {1o, 2o}
4		funsng 6592	. . . 4 ⊢ ((dom 𝐴 ∈ V ∧ 𝑋 ∈ {1o, 2o}) → Fun {⟨dom 𝐴, 𝑋⟩})
5	2, 3, 4	sylancl 598	. . 3 ⊢ (𝐴 ∈ No → Fun {⟨dom 𝐴, 𝑋⟩})
6	3	elexi 3472	. . . . . 6 ⊢ 𝑋 ∈ V
7	6	dmsnop 6217	. . . . 5 ⊢ dom {⟨dom 𝐴, 𝑋⟩} = {dom 𝐴}
8	7	ineq2i 4114	. . . 4 ⊢ (dom 𝐴 ∩ dom {⟨dom 𝐴, 𝑋⟩}) = (dom 𝐴 ∩ {dom 𝐴})
9		nodmord 28016	. . . . . 6 ⊢ (𝐴 ∈ No → Ord dom 𝐴)
10		ordirr 6380	. . . . . 6 ⊢ (Ord dom 𝐴 → ¬ dom 𝐴 ∈ dom 𝐴)
11	9, 10	syl 18	. . . . 5 ⊢ (𝐴 ∈ No → ¬ dom 𝐴 ∈ dom 𝐴)
12		disjsn 4668	. . . . 5 ⊢ ((dom 𝐴 ∩ {dom 𝐴}) = ∅ ↔ ¬ dom 𝐴 ∈ dom 𝐴)
13	11, 12	sylibr 237	. . . 4 ⊢ (𝐴 ∈ No → (dom 𝐴 ∩ {dom 𝐴}) = ∅)
14	8, 13	eqtrid 2807	. . 3 ⊢ (𝐴 ∈ No → (dom 𝐴 ∩ dom {⟨dom 𝐴, 𝑋⟩}) = ∅)
15		funun 6587	. . 3 ⊢ (((Fun 𝐴 ∧ Fun {⟨dom 𝐴, 𝑋⟩}) ∧ (dom 𝐴 ∩ dom {⟨dom 𝐴, 𝑋⟩}) = ∅) → Fun (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}))
16	1, 5, 14, 15	syl21anc 851	. 2 ⊢ (𝐴 ∈ No → Fun (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}))
17	7	uneq2i 4164	. . . 4 ⊢ (dom 𝐴 ∪ dom {⟨dom 𝐴, 𝑋⟩}) = (dom 𝐴 ∪ {dom 𝐴})
18		dmun 5891	. . . 4 ⊢ dom (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) = (dom 𝐴 ∪ dom {⟨dom 𝐴, 𝑋⟩})
19		df-suc 6368	. . . 4 ⊢ suc dom 𝐴 = (dom 𝐴 ∪ {dom 𝐴})
20	17, 18, 19	3eqtr4i 2793	. . 3 ⊢ dom (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) = suc dom 𝐴
21		nodmon 28013	. . . 4 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On)
22		onsuc 7825	. . . 4 ⊢ (dom 𝐴 ∈ On → suc dom 𝐴 ∈ On)
23	21, 22	syl 18	. . 3 ⊢ (𝐴 ∈ No → suc dom 𝐴 ∈ On)
24	20, 23	eqeltrid 2864	. 2 ⊢ (𝐴 ∈ No → dom (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) ∈ On)
25		rnun 6135	. . . 4 ⊢ ran (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) = (ran 𝐴 ∪ ran {⟨dom 𝐴, 𝑋⟩})
26		rnsnopg 6222	. . . . . 6 ⊢ (dom 𝐴 ∈ V → ran {⟨dom 𝐴, 𝑋⟩} = {𝑋})
27	2, 26	syl 18	. . . . 5 ⊢ (𝐴 ∈ No → ran {⟨dom 𝐴, 𝑋⟩} = {𝑋})
28	27	uneq2d 4167	. . . 4 ⊢ (𝐴 ∈ No → (ran 𝐴 ∪ ran {⟨dom 𝐴, 𝑋⟩}) = (ran 𝐴 ∪ {𝑋}))
29	25, 28	eqtrid 2807	. . 3 ⊢ (𝐴 ∈ No → ran (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) = (ran 𝐴 ∪ {𝑋}))
30		norn 28014	. . . 4 ⊢ (𝐴 ∈ No → ran 𝐴 ⊆ {1o, 2o})
31		snssi 4742	. . . . 5 ⊢ (𝑋 ∈ {1o, 2o} → {𝑋} ⊆ {1o, 2o})
32	3, 31	mp1i 14	. . . 4 ⊢ (𝐴 ∈ No → {𝑋} ⊆ {1o, 2o})
33	30, 32	unssd 4190	. . 3 ⊢ (𝐴 ∈ No → (ran 𝐴 ∪ {𝑋}) ⊆ {1o, 2o})
34	29, 33	eqsstrd 3942	. 2 ⊢ (𝐴 ∈ No → ran (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) ⊆ {1o, 2o})
35		elno2 28017	. 2 ⊢ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) ∈ No ↔ (Fun (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) ∧ dom (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) ∈ On ∧ ran (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) ⊆ {1o, 2o}))
36	16, 24, 34, 35	syl3anbrc 1362	1 ⊢ (𝐴 ∈ No → (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) ∈ No)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1569   ∈ wcel 2144  Vcvv 3450   ⊆ wss 3895   ∩ cin 4089   ∪ cun 4149  ∅c0 4278  {csn 4584  {cpr 4586  ⟨cop 4834  dom cdm 5650  ran crn 5651  Ord word 6361  Oncon0 6362  suc csuc 6364  Fun wfun 6533  1_oc1o 8473  2_oc2o 8474  Nocsur 28003

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-ord 6365  df-on 6366  df-suc 6368  df-fun 6541  df-fn 6542  df-f 6543  df-no 28006

This theorem is used by:  noextendlt  28032  noextendgt  28033  nosupno  28066  nosupbnd1  28077  nosupbnd2lem1  28078  noinfno  28081  noinfbnd1  28092  noinfbnd2lem1  28093