Theorem noextendseq 33797

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 33779	. . . 4 ⊢ (𝐴 ∈ No → Fun 𝐴)
2		noextend.1	. . . . 5 ⊢ 𝑋 ∈ {1o, 2o}
3		fnconstg 6646	. . . . 5 ⊢ (𝑋 ∈ {1o, 2o} → ((𝐵 ∖ dom 𝐴) × {𝑋}) Fn (𝐵 ∖ dom 𝐴))
4		fnfun 6517	. . . . 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 4707	. . . . . . . 8 ⊢ (𝑋 ∈ {1o, 2o} → {𝑋} ≠ ∅)
7		dmxp 5827	. . . . . . . 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 4140	. . . . . 6 ⊢ (dom 𝐴 ∩ dom ((𝐵 ∖ dom 𝐴) × {𝑋})) = (dom 𝐴 ∩ (𝐵 ∖ dom 𝐴))
10		disjdif 4402	. . . . . 6 ⊢ (dom 𝐴 ∩ (𝐵 ∖ dom 𝐴)) = ∅
11	9, 10	eqtri 2766	. . . . 5 ⊢ (dom 𝐴 ∩ dom ((𝐵 ∖ dom 𝐴) × {𝑋})) = ∅
12		funun 6464	. . . . 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 687	. . . 4 ⊢ ((Fun 𝐴 ∧ Fun ((𝐵 ∖ dom 𝐴) × {𝑋})) → Fun (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})))
14	1, 5, 13	sylancl 585	. . 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 5808	. . . 4 ⊢ dom (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) = (dom 𝐴 ∪ dom ((𝐵 ∖ dom 𝐴) × {𝑋}))
17	8	uneq2i 4090	. . . 4 ⊢ (dom 𝐴 ∪ dom ((𝐵 ∖ dom 𝐴) × {𝑋})) = (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴))
18	16, 17	eqtri 2766	. . 3 ⊢ dom (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) = (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴))
19		nodmon 33780	. . . 4 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On)
20		undif 4412	. . . . . 6 ⊢ (dom 𝐴 ⊆ 𝐵 ↔ (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) = 𝐵)
21		eleq1a 2834	. . . . . . 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	syl5bi 241	. . . . 5 ⊢ ((dom 𝐴 ∈ On ∧ 𝐵 ∈ On) → (dom 𝐴 ⊆ 𝐵 → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On))
24		ssdif0 4294	. . . . . 6 ⊢ (𝐵 ⊆ dom 𝐴 ↔ (𝐵 ∖ dom 𝐴) = ∅)
25		uneq2 4087	. . . . . . . . . 10 ⊢ ((𝐵 ∖ dom 𝐴) = ∅ → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) = (dom 𝐴 ∪ ∅))
26		un0 4321	. . . . . . . . . 10 ⊢ (dom 𝐴 ∪ ∅) = dom 𝐴
27	25, 26	eqtrdi 2795	. . . . . . . . 9 ⊢ ((𝐵 ∖ dom 𝐴) = ∅ → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) = dom 𝐴)
28	27	eleq1d 2823	. . . . . . . 8 ⊢ ((𝐵 ∖ dom 𝐴) = ∅ → ((dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On ↔ dom 𝐴 ∈ On))
29	28	biimprcd 249	. . . . . . 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	syl5bi 241	. . . . 5 ⊢ ((dom 𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ⊆ dom 𝐴 → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On))
32		eloni 6261	. . . . . 6 ⊢ (dom 𝐴 ∈ On → Ord dom 𝐴)
33		eloni 6261	. . . . . 6 ⊢ (𝐵 ∈ On → Ord 𝐵)
34		ordtri2or2 6347	. . . . . 6 ⊢ ((Ord dom 𝐴 ∧ Ord 𝐵) → (dom 𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ dom 𝐴))
35	32, 33, 34	syl2an 595	. . . . 5 ⊢ ((dom 𝐴 ∈ On ∧ 𝐵 ∈ On) → (dom 𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ dom 𝐴))
36	23, 31, 35	mpjaod 856	. . . 4 ⊢ ((dom 𝐴 ∈ On ∧ 𝐵 ∈ On) → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On)
37	19, 36	sylan 579	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On)
38	18, 37	eqeltrid 2843	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → dom (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) ∈ On)
39		rnun 6038	. . 3 ⊢ ran (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) = (ran 𝐴 ∪ ran ((𝐵 ∖ dom 𝐴) × {𝑋}))
40		norn 33781	. . . . 5 ⊢ (𝐴 ∈ No → ran 𝐴 ⊆ {1o, 2o})
41	40	adantr 480	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → ran 𝐴 ⊆ {1o, 2o})
42		rnxpss 6064	. . . . 5 ⊢ ran ((𝐵 ∖ dom 𝐴) × {𝑋}) ⊆ {𝑋}
43		snssi 4738	. . . . . 6 ⊢ (𝑋 ∈ {1o, 2o} → {𝑋} ⊆ {1o, 2o})
44	2, 43	ax-mp 5	. . . . 5 ⊢ {𝑋} ⊆ {1o, 2o}
45	42, 44	sstri 3926	. . . 4 ⊢ ran ((𝐵 ∖ dom 𝐴) × {𝑋}) ⊆ {1o, 2o}
46		unss 4114	. . . 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 588	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → (ran 𝐴 ∪ ran ((𝐵 ∖ dom 𝐴) × {𝑋})) ⊆ {1o, 2o})
48	39, 47	eqsstrid 3965	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → ran (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) ⊆ {1o, 2o})
49		elno2 33784	. 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 1341	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) ∈ No )

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 843   = wceq 1539   ∈ wcel 2108   ≠ wne 2942   ∖ cdif 3880   ∪ cun 3881   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  {csn 4558  {cpr 4560   × cxp 5578  dom cdm 5580  ran crn 5581  Ord word 6250  Oncon0 6251  Fun wfun 6412   Fn wfn 6413  1_oc1o 8260  2_oc2o 8261   No csur 33770

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-no 33773

This theorem is referenced by:  noetasuplem1  33863  noetainflem1  33867