Theorem noextendseq 33607

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 33589	. . . 4 ⊢ (𝐴 ∈ No → Fun 𝐴)
2		noextend.1	. . . . 5 ⊢ 𝑋 ∈ {1o, 2o}
3		fnconstg 6607	. . . . 5 ⊢ (𝑋 ∈ {1o, 2o} → ((𝐵 ∖ dom 𝐴) × {𝑋}) Fn (𝐵 ∖ dom 𝐴))
4		fnfun 6479	. . . . 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 4690	. . . . . . . 8 ⊢ (𝑋 ∈ {1o, 2o} → {𝑋} ≠ ∅)
7		dmxp 5798	. . . . . . . 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 4124	. . . . . 6 ⊢ (dom 𝐴 ∩ dom ((𝐵 ∖ dom 𝐴) × {𝑋})) = (dom 𝐴 ∩ (𝐵 ∖ dom 𝐴))
10		disjdif 4386	. . . . . 6 ⊢ (dom 𝐴 ∩ (𝐵 ∖ dom 𝐴)) = ∅
11	9, 10	eqtri 2765	. . . . 5 ⊢ (dom 𝐴 ∩ dom ((𝐵 ∖ dom 𝐴) × {𝑋})) = ∅
12		funun 6426	. . . . 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 691	. . . 4 ⊢ ((Fun 𝐴 ∧ Fun ((𝐵 ∖ dom 𝐴) × {𝑋})) → Fun (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})))
14	1, 5, 13	sylancl 589	. . 3 ⊢ (𝐴 ∈ No → Fun (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})))
15	14	adantr 484	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → Fun (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})))
16		dmun 5779	. . . 4 ⊢ dom (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) = (dom 𝐴 ∪ dom ((𝐵 ∖ dom 𝐴) × {𝑋}))
17	8	uneq2i 4074	. . . 4 ⊢ (dom 𝐴 ∪ dom ((𝐵 ∖ dom 𝐴) × {𝑋})) = (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴))
18	16, 17	eqtri 2765	. . 3 ⊢ dom (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) = (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴))
19		nodmon 33590	. . . 4 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On)
20		undif 4396	. . . . . 6 ⊢ (dom 𝐴 ⊆ 𝐵 ↔ (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) = 𝐵)
21		eleq1a 2833	. . . . . . 7 ⊢ (𝐵 ∈ On → ((dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) = 𝐵 → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On))
22	21	adantl 485	. . . . . 6 ⊢ ((dom 𝐴 ∈ On ∧ 𝐵 ∈ On) → ((dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) = 𝐵 → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On))
23	20, 22	syl5bi 245	. . . . 5 ⊢ ((dom 𝐴 ∈ On ∧ 𝐵 ∈ On) → (dom 𝐴 ⊆ 𝐵 → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On))
24		ssdif0 4278	. . . . . 6 ⊢ (𝐵 ⊆ dom 𝐴 ↔ (𝐵 ∖ dom 𝐴) = ∅)
25		uneq2 4071	. . . . . . . . . 10 ⊢ ((𝐵 ∖ dom 𝐴) = ∅ → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) = (dom 𝐴 ∪ ∅))
26		un0 4305	. . . . . . . . . 10 ⊢ (dom 𝐴 ∪ ∅) = dom 𝐴
27	25, 26	eqtrdi 2794	. . . . . . . . 9 ⊢ ((𝐵 ∖ dom 𝐴) = ∅ → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) = dom 𝐴)
28	27	eleq1d 2822	. . . . . . . 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 484	. . . . . 6 ⊢ ((dom 𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 ∖ dom 𝐴) = ∅ → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On))
31	24, 30	syl5bi 245	. . . . 5 ⊢ ((dom 𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ⊆ dom 𝐴 → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On))
32		eloni 6223	. . . . . 6 ⊢ (dom 𝐴 ∈ On → Ord dom 𝐴)
33		eloni 6223	. . . . . 6 ⊢ (𝐵 ∈ On → Ord 𝐵)
34		ordtri2or2 6309	. . . . . 6 ⊢ ((Ord dom 𝐴 ∧ Ord 𝐵) → (dom 𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ dom 𝐴))
35	32, 33, 34	syl2an 599	. . . . 5 ⊢ ((dom 𝐴 ∈ On ∧ 𝐵 ∈ On) → (dom 𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ dom 𝐴))
36	23, 31, 35	mpjaod 860	. . . 4 ⊢ ((dom 𝐴 ∈ On ∧ 𝐵 ∈ On) → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On)
37	19, 36	sylan 583	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On)
38	18, 37	eqeltrid 2842	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → dom (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) ∈ On)
39		rnun 6009	. . 3 ⊢ ran (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) = (ran 𝐴 ∪ ran ((𝐵 ∖ dom 𝐴) × {𝑋}))
40		norn 33591	. . . . 5 ⊢ (𝐴 ∈ No → ran 𝐴 ⊆ {1o, 2o})
41	40	adantr 484	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → ran 𝐴 ⊆ {1o, 2o})
42		rnxpss 6035	. . . . 5 ⊢ ran ((𝐵 ∖ dom 𝐴) × {𝑋}) ⊆ {𝑋}
43		snssi 4721	. . . . . 6 ⊢ (𝑋 ∈ {1o, 2o} → {𝑋} ⊆ {1o, 2o})
44	2, 43	ax-mp 5	. . . . 5 ⊢ {𝑋} ⊆ {1o, 2o}
45	42, 44	sstri 3910	. . . 4 ⊢ ran ((𝐵 ∖ dom 𝐴) × {𝑋}) ⊆ {1o, 2o}
46		unss 4098	. . . 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 592	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → (ran 𝐴 ∪ ran ((𝐵 ∖ dom 𝐴) × {𝑋})) ⊆ {1o, 2o})
48	39, 47	eqsstrid 3949	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → ran (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) ⊆ {1o, 2o})
49		elno2 33594	. 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 1345	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) ∈ No )

Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 847   = wceq 1543   ∈ wcel 2110   ≠ wne 2940   ∖ cdif 3863   ∪ cun 3864   ∩ cin 3865   ⊆ wss 3866  ∅c0 4237  {csn 4541  {cpr 4543   × cxp 5549  dom cdm 5551  ran crn 5552  Ord word 6212  Oncon0 6213  Fun wfun 6374   Fn wfn 6375  1_oc1o 8195  2_oc2o 8196   No csur 33580

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pr 5322

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-ord 6216  df-on 6217  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-no 33583

This theorem is referenced by:  noetasuplem1  33673  noetainflem1  33677