Theorem noextend 33286

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 33269	. . 3 ⊢ (𝐴 ∈ No → Fun 𝐴)
2		dmexg 7594	. . . 4 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ V)
3		noextend.1	. . . 4 ⊢ 𝑋 ∈ {1o, 2o}
4		funsng 6375	. . . 4 ⊢ ((dom 𝐴 ∈ V ∧ 𝑋 ∈ {1o, 2o}) → Fun {⟨dom 𝐴, 𝑋⟩})
5	2, 3, 4	sylancl 589	. . 3 ⊢ (𝐴 ∈ No → Fun {⟨dom 𝐴, 𝑋⟩})
6	3	elexi 3460	. . . . . 6 ⊢ 𝑋 ∈ V
7	6	dmsnop 6040	. . . . 5 ⊢ dom {⟨dom 𝐴, 𝑋⟩} = {dom 𝐴}
8	7	ineq2i 4136	. . . 4 ⊢ (dom 𝐴 ∩ dom {⟨dom 𝐴, 𝑋⟩}) = (dom 𝐴 ∩ {dom 𝐴})
9		nodmord 33273	. . . . . 6 ⊢ (𝐴 ∈ No → Ord dom 𝐴)
10		ordirr 6177	. . . . . 6 ⊢ (Ord dom 𝐴 → ¬ dom 𝐴 ∈ dom 𝐴)
11	9, 10	syl 17	. . . . 5 ⊢ (𝐴 ∈ No → ¬ dom 𝐴 ∈ dom 𝐴)
12		disjsn 4607	. . . . 5 ⊢ ((dom 𝐴 ∩ {dom 𝐴}) = ∅ ↔ ¬ dom 𝐴 ∈ dom 𝐴)
13	11, 12	sylibr 237	. . . 4 ⊢ (𝐴 ∈ No → (dom 𝐴 ∩ {dom 𝐴}) = ∅)
14	8, 13	syl5eq 2845	. . 3 ⊢ (𝐴 ∈ No → (dom 𝐴 ∩ dom {⟨dom 𝐴, 𝑋⟩}) = ∅)
15		funun 6370	. . 3 ⊢ (((Fun 𝐴 ∧ Fun {⟨dom 𝐴, 𝑋⟩}) ∧ (dom 𝐴 ∩ dom {⟨dom 𝐴, 𝑋⟩}) = ∅) → Fun (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}))
16	1, 5, 14, 15	syl21anc 836	. 2 ⊢ (𝐴 ∈ No → Fun (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}))
17	7	uneq2i 4087	. . . 4 ⊢ (dom 𝐴 ∪ dom {⟨dom 𝐴, 𝑋⟩}) = (dom 𝐴 ∪ {dom 𝐴})
18		dmun 5743	. . . 4 ⊢ dom (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) = (dom 𝐴 ∪ dom {⟨dom 𝐴, 𝑋⟩})
19		df-suc 6165	. . . 4 ⊢ suc dom 𝐴 = (dom 𝐴 ∪ {dom 𝐴})
20	17, 18, 19	3eqtr4i 2831	. . 3 ⊢ dom (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) = suc dom 𝐴
21		nodmon 33270	. . . 4 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On)
22		suceloni 7508	. . . 4 ⊢ (dom 𝐴 ∈ On → suc dom 𝐴 ∈ On)
23	21, 22	syl 17	. . 3 ⊢ (𝐴 ∈ No → suc dom 𝐴 ∈ On)
24	20, 23	eqeltrid 2894	. 2 ⊢ (𝐴 ∈ No → dom (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) ∈ On)
25		rnun 5971	. . . 4 ⊢ ran (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) = (ran 𝐴 ∪ ran {⟨dom 𝐴, 𝑋⟩})
26		rnsnopg 6045	. . . . . 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 4090	. . . 4 ⊢ (𝐴 ∈ No → (ran 𝐴 ∪ ran {⟨dom 𝐴, 𝑋⟩}) = (ran 𝐴 ∪ {𝑋}))
29	25, 28	syl5eq 2845	. . 3 ⊢ (𝐴 ∈ No → ran (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) = (ran 𝐴 ∪ {𝑋}))
30		norn 33271	. . . 4 ⊢ (𝐴 ∈ No → ran 𝐴 ⊆ {1o, 2o})
31		snssi 4701	. . . . 5 ⊢ (𝑋 ∈ {1o, 2o} → {𝑋} ⊆ {1o, 2o})
32	3, 31	mp1i 13	. . . 4 ⊢ (𝐴 ∈ No → {𝑋} ⊆ {1o, 2o})
33	30, 32	unssd 4113	. . 3 ⊢ (𝐴 ∈ No → (ran 𝐴 ∪ {𝑋}) ⊆ {1o, 2o})
34	29, 33	eqsstrd 3953	. 2 ⊢ (𝐴 ∈ No → ran (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) ⊆ {1o, 2o})
35		elno2 33274	. 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 1340	1 ⊢ (𝐴 ∈ No → (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) ∈ No )

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1538   ∈ wcel 2111  Vcvv 3441   ∪ cun 3879   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243  {csn 4525  {cpr 4527  ⟨cop 4531  dom cdm 5519  ran crn 5520  Ord word 6158  Oncon0 6159  suc csuc 6161  Fun wfun 6318  1_oc1o 8078  2_oc2o 8079   No csur 33260

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6162  df-on 6163  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-no 33263

This theorem is referenced by:  noextendlt  33289  noextendgt  33290  nosupno  33316  nosupbnd1  33327  nosupbnd2lem1  33328