Theorem noextend 27859

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 27842	. . 3 ⊢ (𝐴 ∈ No → Fun 𝐴)
2		dmexg 7900	. . . 4 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ V)
3		noextend.1	. . . 4 ⊢ 𝑋 ∈ {1o, 2o}
4		funsng 6591	. . . 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 3479	. . . . . 6 ⊢ 𝑋 ∈ V
7	6	dmsnop 6219	. . . . 5 ⊢ dom {⟨dom 𝐴, 𝑋⟩} = {dom 𝐴}
8	7	ineq2i 4170	. . . 4 ⊢ (dom 𝐴 ∩ dom {⟨dom 𝐴, 𝑋⟩}) = (dom 𝐴 ∩ {dom 𝐴})
9		nodmord 27846	. . . . . 6 ⊢ (𝐴 ∈ No → Ord dom 𝐴)
10		ordirr 6382	. . . . . 6 ⊢ (Ord dom 𝐴 → ¬ dom 𝐴 ∈ dom 𝐴)
11	9, 10	syl 18	. . . . 5 ⊢ (𝐴 ∈ No → ¬ dom 𝐴 ∈ dom 𝐴)
12		disjsn 4679	. . . . 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 2812	. . 3 ⊢ (𝐴 ∈ No → (dom 𝐴 ∩ dom {⟨dom 𝐴, 𝑋⟩}) = ∅)
15		funun 6586	. . 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 4119	. . . 4 ⊢ (dom 𝐴 ∪ dom {⟨dom 𝐴, 𝑋⟩}) = (dom 𝐴 ∪ {dom 𝐴})
18		dmun 5902	. . . 4 ⊢ dom (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) = (dom 𝐴 ∪ dom {⟨dom 𝐴, 𝑋⟩})
19		df-suc 6370	. . . 4 ⊢ suc dom 𝐴 = (dom 𝐴 ∪ {dom 𝐴})
20	17, 18, 19	3eqtr4i 2798	. . 3 ⊢ dom (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) = suc dom 𝐴
21		nodmon 27843	. . . 4 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On)
22		onsuc 7811	. . . 4 ⊢ (dom 𝐴 ∈ On → suc dom 𝐴 ∈ On)
23	21, 22	syl 18	. . 3 ⊢ (𝐴 ∈ No → suc dom 𝐴 ∈ On)
24	20, 23	eqeltrid 2869	. 2 ⊢ (𝐴 ∈ No → dom (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) ∈ On)
25		rnun 6144	. . . 4 ⊢ ran (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) = (ran 𝐴 ∪ ran {⟨dom 𝐴, 𝑋⟩})
26		rnsnopg 6224	. . . . . 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 4122	. . . 4 ⊢ (𝐴 ∈ No → (ran 𝐴 ∪ ran {⟨dom 𝐴, 𝑋⟩}) = (ran 𝐴 ∪ {𝑋}))
29	25, 28	eqtrid 2812	. . 3 ⊢ (𝐴 ∈ No → ran (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) = (ran 𝐴 ∪ {𝑋}))
30		norn 27844	. . . 4 ⊢ (𝐴 ∈ No → ran 𝐴 ⊆ {1o, 2o})
31		snssi 4753	. . . . 5 ⊢ (𝑋 ∈ {1o, 2o} → {𝑋} ⊆ {1o, 2o})
32	3, 31	mp1i 14	. . . 4 ⊢ (𝐴 ∈ No → {𝑋} ⊆ {1o, 2o})
33	30, 32	unssd 4145	. . 3 ⊢ (𝐴 ∈ No → (ran 𝐴 ∪ {𝑋}) ⊆ {1o, 2o})
34	29, 33	eqsstrd 3972	. 2 ⊢ (𝐴 ∈ No → ran (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) ⊆ {1o, 2o})
35		elno2 27847	. 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 1570   ∈ wcel 2146  Vcvv 3457   ∪ cun 3904   ∩ cin 3905   ⊆ wss 3906  ∅c0 4286  {csn 4591  {cpr 4593  ⟨cop 4597  dom cdm 5663  ran crn 5664  Ord word 6363  Oncon0 6364  suc csuc 6366  Fun wfun 6534  1_oc1o 8448  2_oc2o 8449   No csur 27833

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pow 5338  ax-pr 5406  ax-un 7738

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-sb 2100  df-mo 2569  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-ord 6367  df-on 6368  df-suc 6370  df-fun 6542  df-fn 6543  df-f 6544  df-no 27836

This theorem is used by:  noextendlt  27862  noextendgt  27863  nosupno  27896  nosupbnd1  27907  nosupbnd2lem1  27908  noinfno  27911  noinfbnd1  27922  noinfbnd2lem1  27923