Theorem noextend 27634

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 27617	. . 3 ⊢ (𝐴 ∈ No → Fun 𝐴)
2		dmexg 7843	. . . 4 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ V)
3		noextend.1	. . . 4 ⊢ 𝑋 ∈ {1o, 2o}
4		funsng 6543	. . . 4 ⊢ ((dom 𝐴 ∈ V ∧ 𝑋 ∈ {1o, 2o}) → Fun {⟨dom 𝐴, 𝑋⟩})
5	2, 3, 4	sylancl 586	. . 3 ⊢ (𝐴 ∈ No → Fun {⟨dom 𝐴, 𝑋⟩})
6	3	elexi 3463	. . . . . 6 ⊢ 𝑋 ∈ V
7	6	dmsnop 6174	. . . . 5 ⊢ dom {⟨dom 𝐴, 𝑋⟩} = {dom 𝐴}
8	7	ineq2i 4169	. . . 4 ⊢ (dom 𝐴 ∩ dom {⟨dom 𝐴, 𝑋⟩}) = (dom 𝐴 ∩ {dom 𝐴})
9		nodmord 27621	. . . . . 6 ⊢ (𝐴 ∈ No → Ord dom 𝐴)
10		ordirr 6335	. . . . . 6 ⊢ (Ord dom 𝐴 → ¬ dom 𝐴 ∈ dom 𝐴)
11	9, 10	syl 17	. . . . 5 ⊢ (𝐴 ∈ No → ¬ dom 𝐴 ∈ dom 𝐴)
12		disjsn 4668	. . . . 5 ⊢ ((dom 𝐴 ∩ {dom 𝐴}) = ∅ ↔ ¬ dom 𝐴 ∈ dom 𝐴)
13	11, 12	sylibr 234	. . . 4 ⊢ (𝐴 ∈ No → (dom 𝐴 ∩ {dom 𝐴}) = ∅)
14	8, 13	eqtrid 2783	. . 3 ⊢ (𝐴 ∈ No → (dom 𝐴 ∩ dom {⟨dom 𝐴, 𝑋⟩}) = ∅)
15		funun 6538	. . 3 ⊢ (((Fun 𝐴 ∧ Fun {⟨dom 𝐴, 𝑋⟩}) ∧ (dom 𝐴 ∩ dom {⟨dom 𝐴, 𝑋⟩}) = ∅) → Fun (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}))
16	1, 5, 14, 15	syl21anc 837	. 2 ⊢ (𝐴 ∈ No → Fun (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}))
17	7	uneq2i 4117	. . . 4 ⊢ (dom 𝐴 ∪ dom {⟨dom 𝐴, 𝑋⟩}) = (dom 𝐴 ∪ {dom 𝐴})
18		dmun 5859	. . . 4 ⊢ dom (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) = (dom 𝐴 ∪ dom {⟨dom 𝐴, 𝑋⟩})
19		df-suc 6323	. . . 4 ⊢ suc dom 𝐴 = (dom 𝐴 ∪ {dom 𝐴})
20	17, 18, 19	3eqtr4i 2769	. . 3 ⊢ dom (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) = suc dom 𝐴
21		nodmon 27618	. . . 4 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On)
22		onsuc 7755	. . . 4 ⊢ (dom 𝐴 ∈ On → suc dom 𝐴 ∈ On)
23	21, 22	syl 17	. . 3 ⊢ (𝐴 ∈ No → suc dom 𝐴 ∈ On)
24	20, 23	eqeltrid 2840	. 2 ⊢ (𝐴 ∈ No → dom (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) ∈ On)
25		rnun 6103	. . . 4 ⊢ ran (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) = (ran 𝐴 ∪ ran {⟨dom 𝐴, 𝑋⟩})
26		rnsnopg 6179	. . . . . 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 4120	. . . 4 ⊢ (𝐴 ∈ No → (ran 𝐴 ∪ ran {⟨dom 𝐴, 𝑋⟩}) = (ran 𝐴 ∪ {𝑋}))
29	25, 28	eqtrid 2783	. . 3 ⊢ (𝐴 ∈ No → ran (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) = (ran 𝐴 ∪ {𝑋}))
30		norn 27619	. . . 4 ⊢ (𝐴 ∈ No → ran 𝐴 ⊆ {1o, 2o})
31		snssi 4764	. . . . 5 ⊢ (𝑋 ∈ {1o, 2o} → {𝑋} ⊆ {1o, 2o})
32	3, 31	mp1i 13	. . . 4 ⊢ (𝐴 ∈ No → {𝑋} ⊆ {1o, 2o})
33	30, 32	unssd 4144	. . 3 ⊢ (𝐴 ∈ No → (ran 𝐴 ∪ {𝑋}) ⊆ {1o, 2o})
34	29, 33	eqsstrd 3968	. 2 ⊢ (𝐴 ∈ No → ran (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) ⊆ {1o, 2o})
35		elno2 27622	. 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 1344	1 ⊢ (𝐴 ∈ No → (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) ∈ No )

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1541   ∈ wcel 2113  Vcvv 3440   ∪ cun 3899   ∩ cin 3900   ⊆ wss 3901  ∅c0 4285  {csn 4580  {cpr 4582  ⟨cop 4586  dom cdm 5624  ran crn 5625  Ord word 6316  Oncon0 6317  suc csuc 6319  Fun wfun 6486  1_oc1o 8390  2_oc2o 8391   No csur 27607

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 2115  ax-9 2123  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

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-sb 2068  df-mo 2539  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-ord 6320  df-on 6321  df-suc 6323  df-fun 6494  df-fn 6495  df-f 6496  df-no 27610

This theorem is referenced by:  noextendlt  27637  noextendgt  27638  nosupno  27671  nosupbnd1  27682  nosupbnd2lem1  27683  noinfno  27686  noinfbnd1  27697  noinfbnd2lem1  27698