Theorem noextendseq 28054

Description: Extend a surreal by a sequence of ordinals. (Contributed by Scott Fenton, 30-Nov-2021.)

Hypothesis

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

Assertion

Ref	Expression
noextendseq	⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) ∈ No)

Proof of Theorem noextendseq

Step	Hyp	Ref	Expression
1		nofun 28036	. . . 4 ⊢ (𝐴 ∈ No → Fun 𝐴)
2		noextend.1	. . . . 5 ⊢ 𝑋 ∈ {1o, 2o}
3		fnconstg 6792	. . . . 5 ⊢ (𝑋 ∈ {1o, 2o} → ((𝐵 ∖ dom 𝐴) × {𝑋}) Fn (𝐵 ∖ dom 𝐴))
4		fnfun 6661	. . . . 5 ⊢ (((𝐵 ∖ dom 𝐴) × {𝑋}) Fn (𝐵 ∖ dom 𝐴) → Fun ((𝐵 ∖ dom 𝐴) × {𝑋}))
5	2, 3, 4	mp2b 10	. . . 4 ⊢ Fun ((𝐵 ∖ dom 𝐴) × {𝑋})
6		snnzg 4735	. . . . . . . 8 ⊢ (𝑋 ∈ {1o, 2o} → {𝑋} ≠ ∅)
7		dmxp 5918	. . . . . . . 8 ⊢ ({𝑋} ≠ ∅ → dom ((𝐵 ∖ dom 𝐴) × {𝑋}) = (𝐵 ∖ dom 𝐴))
8	2, 6, 7	mp2b 10	. . . . . . 7 ⊢ dom ((𝐵 ∖ dom 𝐴) × {𝑋}) = (𝐵 ∖ dom 𝐴)
9	8	ineq2i 4117	. . . . . 6 ⊢ (dom 𝐴 ∩ dom ((𝐵 ∖ dom 𝐴) × {𝑋})) = (dom 𝐴 ∩ (𝐵 ∖ dom 𝐴))
10		disjdif 4429	. . . . . 6 ⊢ (dom 𝐴 ∩ (𝐵 ∖ dom 𝐴)) = ∅
11	9, 10	eqtri 2786	. . . . 5 ⊢ (dom 𝐴 ∩ dom ((𝐵 ∖ dom 𝐴) × {𝑋})) = ∅
12		funun 6606	. . . . 5 ⊢ (((Fun 𝐴 ∧ Fun ((𝐵 ∖ dom 𝐴) × {𝑋})) ∧ (dom 𝐴 ∩ dom ((𝐵 ∖ dom 𝐴) × {𝑋})) = ∅) → Fun (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})))
13	11, 12	mpan2 704	. . . 4 ⊢ ((Fun 𝐴 ∧ Fun ((𝐵 ∖ dom 𝐴) × {𝑋})) → Fun (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})))
14	1, 5, 13	sylancl 598	. . 3 ⊢ (𝐴 ∈ No → Fun (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})))
15	14	adantr 486	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → Fun (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})))
16		dmun 5899	. . . 4 ⊢ dom (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) = (dom 𝐴 ∪ dom ((𝐵 ∖ dom 𝐴) × {𝑋}))
17	8	uneq2i 4167	. . . 4 ⊢ (dom 𝐴 ∪ dom ((𝐵 ∖ dom 𝐴) × {𝑋})) = (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴))
18	16, 17	eqtri 2786	. . 3 ⊢ dom (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) = (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴))
19		nodmon 28037	. . . 4 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On)
20		undif 4441	. . . . . 6 ⊢ (dom 𝐴 ⊆ 𝐵 ↔ (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) = 𝐵)
21		eleq1a 2859	. . . . . . 7 ⊢ (𝐵 ∈ On → ((dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) = 𝐵 → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On))
22	21	adantl 487	. . . . . 6 ⊢ ((dom 𝐴 ∈ On ∧ 𝐵 ∈ On) → ((dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) = 𝐵 → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On))
23	20, 22	biimtrid 245	. . . . 5 ⊢ ((dom 𝐴 ∈ On ∧ 𝐵 ∈ On) → (dom 𝐴 ⊆ 𝐵 → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On))
24		ssdif0 4317	. . . . . 6 ⊢ (𝐵 ⊆ dom 𝐴 ↔ (𝐵 ∖ dom 𝐴) = ∅)
25		uneq2 4164	. . . . . . . . . 10 ⊢ ((𝐵 ∖ dom 𝐴) = ∅ → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) = (dom 𝐴 ∪ ∅))
26		un0 4347	. . . . . . . . . 10 ⊢ (dom 𝐴 ∪ ∅) = dom 𝐴
27	25, 26	eqtrdi 2814	. . . . . . . . 9 ⊢ ((𝐵 ∖ dom 𝐴) = ∅ → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) = dom 𝐴)
28	27	eleq1d 2849	. . . . . . . 8 ⊢ ((𝐵 ∖ dom 𝐴) = ∅ → ((dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On ↔ dom 𝐴 ∈ On))
29	28	biimprcd 253	. . . . . . 7 ⊢ (dom 𝐴 ∈ On → ((𝐵 ∖ dom 𝐴) = ∅ → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On))
30	29	adantr 486	. . . . . 6 ⊢ ((dom 𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 ∖ dom 𝐴) = ∅ → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On))
31	24, 30	biimtrid 245	. . . . 5 ⊢ ((dom 𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ⊆ dom 𝐴 → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On))
32		eloni 6390	. . . . . 6 ⊢ (dom 𝐴 ∈ On → Ord dom 𝐴)
33		eloni 6390	. . . . . 6 ⊢ (𝐵 ∈ On → Ord 𝐵)
34		ordtri2or2 6482	. . . . . 6 ⊢ ((Ord dom 𝐴 ∧ Ord 𝐵) → (dom 𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ dom 𝐴))
35	32, 33, 34	syl2an 608	. . . . 5 ⊢ ((dom 𝐴 ∈ On ∧ 𝐵 ∈ On) → (dom 𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ dom 𝐴))
36	23, 31, 35	mpjaod 874	. . . 4 ⊢ ((dom 𝐴 ∈ On ∧ 𝐵 ∈ On) → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On)
37	19, 36	sylan 592	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On)
38	18, 37	eqeltrid 2868	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → dom (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) ∈ On)
39		rnun 6148	. . 3 ⊢ ran (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) = (ran 𝐴 ∪ ran ((𝐵 ∖ dom 𝐴) × {𝑋}))
40		norn 28038	. . . . 5 ⊢ (𝐴 ∈ No → ran 𝐴 ⊆ {1o, 2o})
41	40	adantr 486	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → ran 𝐴 ⊆ {1o, 2o})
42		rnxpss 6176	. . . . 5 ⊢ ran ((𝐵 ∖ dom 𝐴) × {𝑋}) ⊆ {𝑋}
43		snssi 4746	. . . . . 6 ⊢ (𝑋 ∈ {1o, 2o} → {𝑋} ⊆ {1o, 2o})
44	2, 43	ax-mp 5	. . . . 5 ⊢ {𝑋} ⊆ {1o, 2o}
45	42, 44	sstri 3922	. . . 4 ⊢ ran ((𝐵 ∖ dom 𝐴) × {𝑋}) ⊆ {1o, 2o}
46		unss 4191	. . . 4 ⊢ ((ran 𝐴 ⊆ {1o, 2o} ∧ ran ((𝐵 ∖ dom 𝐴) × {𝑋}) ⊆ {1o, 2o}) ↔ (ran 𝐴 ∪ ran ((𝐵 ∖ dom 𝐴) × {𝑋})) ⊆ {1o, 2o})
47	41, 45, 46	sylanblc 601	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → (ran 𝐴 ∪ ran ((𝐵 ∖ dom 𝐴) × {𝑋})) ⊆ {1o, 2o})
48	39, 47	eqsstrid 3951	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → ran (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) ⊆ {1o, 2o})
49		elno2 28041	. 2 ⊢ ((𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) ∈ No ↔ (Fun (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) ∧ dom (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) ∈ On ∧ ran (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) ⊆ {1o, 2o}))
50	15, 38, 48, 49	syl3anbrc 1362	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) ∈ No)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147   ≠ wne 2959   ⊆ wss 3900   ∖ cdif 4048   ∩ cin 4092   ∪ cun 4152  ∅c0 4281  {csn 4588  {cpr 4590   × cxp 5653  dom cdm 5655  ran crn 5656  Ord word 6379  Oncon0 6380  Fun wfun 6552   Fn wfn 6553  1_oc1o 8497  2_oc2o 8498  Nocsur 28027

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-pow 5331  ax-pr 5395  ax-un 7775

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-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-ord 6383  df-on 6384  df-fun 6560  df-fn 6561  df-f 6562  df-no 28030

This theorem is used by:  noetasuplem1  28120  noetainflem1  28124