Theorem noextendseq 27586

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 27568	. . . 4 ⊢ (𝐴 ∈ No → Fun 𝐴)
2		noextend.1	. . . . 5 ⊢ 𝑋 ∈ {1o, 2o}
3		fnconstg 6751	. . . . 5 ⊢ (𝑋 ∈ {1o, 2o} → ((𝐵 ∖ dom 𝐴) × {𝑋}) Fn (𝐵 ∖ dom 𝐴))
4		fnfun 6621	. . . . 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 4741	. . . . . . . 8 ⊢ (𝑋 ∈ {1o, 2o} → {𝑋} ≠ ∅)
7		dmxp 5895	. . . . . . . 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 4183	. . . . . 6 ⊢ (dom 𝐴 ∩ dom ((𝐵 ∖ dom 𝐴) × {𝑋})) = (dom 𝐴 ∩ (𝐵 ∖ dom 𝐴))
10		disjdif 4438	. . . . . 6 ⊢ (dom 𝐴 ∩ (𝐵 ∖ dom 𝐴)) = ∅
11	9, 10	eqtri 2753	. . . . 5 ⊢ (dom 𝐴 ∩ dom ((𝐵 ∖ dom 𝐴) × {𝑋})) = ∅
12		funun 6565	. . . . 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 586	. . 3 ⊢ (𝐴 ∈ No → Fun (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})))
15	14	adantr 480	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → Fun (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})))
16		dmun 5877	. . . 4 ⊢ dom (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) = (dom 𝐴 ∪ dom ((𝐵 ∖ dom 𝐴) × {𝑋}))
17	8	uneq2i 4131	. . . 4 ⊢ (dom 𝐴 ∪ dom ((𝐵 ∖ dom 𝐴) × {𝑋})) = (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴))
18	16, 17	eqtri 2753	. . 3 ⊢ dom (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) = (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴))
19		nodmon 27569	. . . 4 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On)
20		undif 4448	. . . . . 6 ⊢ (dom 𝐴 ⊆ 𝐵 ↔ (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) = 𝐵)
21		eleq1a 2824	. . . . . . 7 ⊢ (𝐵 ∈ On → ((dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) = 𝐵 → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On))
22	21	adantl 481	. . . . . 6 ⊢ ((dom 𝐴 ∈ On ∧ 𝐵 ∈ On) → ((dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) = 𝐵 → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On))
23	20, 22	biimtrid 242	. . . . 5 ⊢ ((dom 𝐴 ∈ On ∧ 𝐵 ∈ On) → (dom 𝐴 ⊆ 𝐵 → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On))
24		ssdif0 4332	. . . . . 6 ⊢ (𝐵 ⊆ dom 𝐴 ↔ (𝐵 ∖ dom 𝐴) = ∅)
25		uneq2 4128	. . . . . . . . . 10 ⊢ ((𝐵 ∖ dom 𝐴) = ∅ → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) = (dom 𝐴 ∪ ∅))
26		un0 4360	. . . . . . . . . 10 ⊢ (dom 𝐴 ∪ ∅) = dom 𝐴
27	25, 26	eqtrdi 2781	. . . . . . . . 9 ⊢ ((𝐵 ∖ dom 𝐴) = ∅ → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) = dom 𝐴)
28	27	eleq1d 2814	. . . . . . . 8 ⊢ ((𝐵 ∖ dom 𝐴) = ∅ → ((dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On ↔ dom 𝐴 ∈ On))
29	28	biimprcd 250	. . . . . . 7 ⊢ (dom 𝐴 ∈ On → ((𝐵 ∖ dom 𝐴) = ∅ → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On))
30	29	adantr 480	. . . . . 6 ⊢ ((dom 𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 ∖ dom 𝐴) = ∅ → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On))
31	24, 30	biimtrid 242	. . . . 5 ⊢ ((dom 𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ⊆ dom 𝐴 → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On))
32		eloni 6345	. . . . . 6 ⊢ (dom 𝐴 ∈ On → Ord dom 𝐴)
33		eloni 6345	. . . . . 6 ⊢ (𝐵 ∈ On → Ord 𝐵)
34		ordtri2or2 6436	. . . . . 6 ⊢ ((Ord dom 𝐴 ∧ Ord 𝐵) → (dom 𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ dom 𝐴))
35	32, 33, 34	syl2an 596	. . . . 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 580	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On)
38	18, 37	eqeltrid 2833	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → dom (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) ∈ On)
39		rnun 6121	. . 3 ⊢ ran (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) = (ran 𝐴 ∪ ran ((𝐵 ∖ dom 𝐴) × {𝑋}))
40		norn 27570	. . . . 5 ⊢ (𝐴 ∈ No → ran 𝐴 ⊆ {1o, 2o})
41	40	adantr 480	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → ran 𝐴 ⊆ {1o, 2o})
42		rnxpss 6148	. . . . 5 ⊢ ran ((𝐵 ∖ dom 𝐴) × {𝑋}) ⊆ {𝑋}
43		snssi 4775	. . . . . 6 ⊢ (𝑋 ∈ {1o, 2o} → {𝑋} ⊆ {1o, 2o})
44	2, 43	ax-mp 5	. . . . 5 ⊢ {𝑋} ⊆ {1o, 2o}
45	42, 44	sstri 3959	. . . 4 ⊢ ran ((𝐵 ∖ dom 𝐴) × {𝑋}) ⊆ {1o, 2o}
46		unss 4156	. . . 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 589	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → (ran 𝐴 ∪ ran ((𝐵 ∖ dom 𝐴) × {𝑋})) ⊆ {1o, 2o})
48	39, 47	eqsstrid 3988	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → ran (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) ⊆ {1o, 2o})
49		elno2 27573	. 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 1344	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) ∈ No )

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109   ≠ wne 2926   ∖ cdif 3914   ∪ cun 3915   ∩ cin 3916   ⊆ wss 3917  ∅c0 4299  {csn 4592  {cpr 4594   × cxp 5639  dom cdm 5641  ran crn 5642  Ord word 6334  Oncon0 6335  Fun wfun 6508   Fn wfn 6509  1_oc1o 8430  2_oc2o 8431   No csur 27558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-ord 6338  df-on 6339  df-fun 6516  df-fn 6517  df-f 6518  df-no 27561

This theorem is referenced by:  noetasuplem1  27652  noetainflem1  27656