Theorem noextend 27648

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 27631	. . 3 ⊢ (𝐴 ∈ No → Fun 𝐴)
2		dmexg 7841	. . . 4 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ V)
3		noextend.1	. . . 4 ⊢ 𝑋 ∈ {1o, 2o}
4		funsng 6536	. . . 4 ⊢ ((dom 𝐴 ∈ V ∧ 𝑋 ∈ {1o, 2o}) → Fun {⟨dom 𝐴, 𝑋⟩})
5	2, 3, 4	sylancl 592	. . 3 ⊢ (𝐴 ∈ No → Fun {⟨dom 𝐴, 𝑋⟩})
6	3	elexi 3453	. . . . . 6 ⊢ 𝑋 ∈ V
7	6	dmsnop 6167	. . . . 5 ⊢ dom {⟨dom 𝐴, 𝑋⟩} = {dom 𝐴}
8	7	ineq2i 4146	. . . 4 ⊢ (dom 𝐴 ∩ dom {⟨dom 𝐴, 𝑋⟩}) = (dom 𝐴 ∩ {dom 𝐴})
9		nodmord 27635	. . . . . 6 ⊢ (𝐴 ∈ No → Ord dom 𝐴)
10		ordirr 6328	. . . . . 6 ⊢ (Ord dom 𝐴 → ¬ dom 𝐴 ∈ dom 𝐴)
11	9, 10	syl 17	. . . . 5 ⊢ (𝐴 ∈ No → ¬ dom 𝐴 ∈ dom 𝐴)
12		disjsn 4643	. . . . 5 ⊢ ((dom 𝐴 ∩ {dom 𝐴}) = ∅ ↔ ¬ dom 𝐴 ∈ dom 𝐴)
13	11, 12	sylibr 235	. . . 4 ⊢ (𝐴 ∈ No → (dom 𝐴 ∩ {dom 𝐴}) = ∅)
14	8, 13	eqtrid 2786	. . 3 ⊢ (𝐴 ∈ No → (dom 𝐴 ∩ dom {⟨dom 𝐴, 𝑋⟩}) = ∅)
15		funun 6531	. . 3 ⊢ (((Fun 𝐴 ∧ Fun {⟨dom 𝐴, 𝑋⟩}) ∧ (dom 𝐴 ∩ dom {⟨dom 𝐴, 𝑋⟩}) = ∅) → Fun (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}))
16	1, 5, 14, 15	syl21anc 843	. 2 ⊢ (𝐴 ∈ No → Fun (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}))
17	7	uneq2i 4095	. . . 4 ⊢ (dom 𝐴 ∪ dom {⟨dom 𝐴, 𝑋⟩}) = (dom 𝐴 ∪ {dom 𝐴})
18		dmun 5852	. . . 4 ⊢ dom (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) = (dom 𝐴 ∪ dom {⟨dom 𝐴, 𝑋⟩})
19		df-suc 6316	. . . 4 ⊢ suc dom 𝐴 = (dom 𝐴 ∪ {dom 𝐴})
20	17, 18, 19	3eqtr4i 2772	. . 3 ⊢ dom (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) = suc dom 𝐴
21		nodmon 27632	. . . 4 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On)
22		onsuc 7753	. . . 4 ⊢ (dom 𝐴 ∈ On → suc dom 𝐴 ∈ On)
23	21, 22	syl 17	. . 3 ⊢ (𝐴 ∈ No → suc dom 𝐴 ∈ On)
24	20, 23	eqeltrid 2843	. 2 ⊢ (𝐴 ∈ No → dom (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) ∈ On)
25		rnun 6096	. . . 4 ⊢ ran (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) = (ran 𝐴 ∪ ran {⟨dom 𝐴, 𝑋⟩})
26		rnsnopg 6172	. . . . . 6 ⊢ (dom 𝐴 ∈ V → ran {⟨dom 𝐴, 𝑋⟩} = {𝑋})
27	2, 26	syl 17	. . . . 5 ⊢ (𝐴 ∈ No → ran {⟨dom 𝐴, 𝑋⟩} = {𝑋})
28	27	uneq2d 4098	. . . 4 ⊢ (𝐴 ∈ No → (ran 𝐴 ∪ ran {⟨dom 𝐴, 𝑋⟩}) = (ran 𝐴 ∪ {𝑋}))
29	25, 28	eqtrid 2786	. . 3 ⊢ (𝐴 ∈ No → ran (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) = (ran 𝐴 ∪ {𝑋}))
30		norn 27633	. . . 4 ⊢ (𝐴 ∈ No → ran 𝐴 ⊆ {1o, 2o})
31		snssi 4717	. . . . 5 ⊢ (𝑋 ∈ {1o, 2o} → {𝑋} ⊆ {1o, 2o})
32	3, 31	mp1i 13	. . . 4 ⊢ (𝐴 ∈ No → {𝑋} ⊆ {1o, 2o})
33	30, 32	unssd 4121	. . 3 ⊢ (𝐴 ∈ No → (ran 𝐴 ∪ {𝑋}) ⊆ {1o, 2o})
34	29, 33	eqsstrd 3949	. 2 ⊢ (𝐴 ∈ No → ran (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) ⊆ {1o, 2o})
35		elno2 27636	. 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 1350	1 ⊢ (𝐴 ∈ No → (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) ∈ No )

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1547   ∈ wcel 2119  Vcvv 3431   ∪ cun 3881   ∩ cin 3882   ⊆ wss 3883  ∅c0 4261  {csn 4555  {cpr 4557  ⟨cop 4561  dom cdm 5618  ran crn 5619  Ord word 6309  Oncon0 6310  suc csuc 6312  Fun wfun 6479  1_oc1o 8388  2_oc2o 8389   No csur 27621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-mo 2543  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-ord 6313  df-on 6314  df-suc 6316  df-fun 6487  df-fn 6488  df-f 6489  df-no 27624

This theorem is referenced by:  noextendlt  27651  noextendgt  27652  nosupno  27685  nosupbnd1  27696  nosupbnd2lem1  27697  noinfno  27700  noinfbnd1  27711  noinfbnd2lem1  27712