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Theorem noextendseq 31945
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
StepHypRef Expression
1 nofun 31927 . . . 4 (𝐴 No → Fun 𝐴)
2 noextend.1 . . . . . 6 𝑋 ∈ {1𝑜, 2𝑜}
3 fnconstg 6131 . . . . . 6 (𝑋 ∈ {1𝑜, 2𝑜} → ((𝐵 ∖ dom 𝐴) × {𝑋}) Fn (𝐵 ∖ dom 𝐴))
4 fnfun 6026 . . . . . 6 (((𝐵 ∖ dom 𝐴) × {𝑋}) Fn (𝐵 ∖ dom 𝐴) → Fun ((𝐵 ∖ dom 𝐴) × {𝑋}))
52, 3, 4mp2b 10 . . . . 5 Fun ((𝐵 ∖ dom 𝐴) × {𝑋})
6 snnzg 4339 . . . . . . . . 9 (𝑋 ∈ {1𝑜, 2𝑜} → {𝑋} ≠ ∅)
7 dmxp 5376 . . . . . . . . 9 ({𝑋} ≠ ∅ → dom ((𝐵 ∖ dom 𝐴) × {𝑋}) = (𝐵 ∖ dom 𝐴))
82, 6, 7mp2b 10 . . . . . . . 8 dom ((𝐵 ∖ dom 𝐴) × {𝑋}) = (𝐵 ∖ dom 𝐴)
98ineq2i 3844 . . . . . . 7 (dom 𝐴 ∩ dom ((𝐵 ∖ dom 𝐴) × {𝑋})) = (dom 𝐴 ∩ (𝐵 ∖ dom 𝐴))
10 disjdif 4073 . . . . . . 7 (dom 𝐴 ∩ (𝐵 ∖ dom 𝐴)) = ∅
119, 10eqtri 2673 . . . . . 6 (dom 𝐴 ∩ dom ((𝐵 ∖ dom 𝐴) × {𝑋})) = ∅
12 funun 5970 . . . . . 6 (((Fun 𝐴 ∧ Fun ((𝐵 ∖ dom 𝐴) × {𝑋})) ∧ (dom 𝐴 ∩ dom ((𝐵 ∖ dom 𝐴) × {𝑋})) = ∅) → Fun (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})))
1311, 12mpan2 707 . . . . 5 ((Fun 𝐴 ∧ Fun ((𝐵 ∖ dom 𝐴) × {𝑋})) → Fun (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})))
145, 13mpan2 707 . . . 4 (Fun 𝐴 → Fun (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})))
151, 14syl 17 . . 3 (𝐴 No → Fun (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})))
1615adantr 480 . 2 ((𝐴 No 𝐵 ∈ On) → Fun (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})))
17 dmun 5363 . . . 4 dom (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) = (dom 𝐴 ∪ dom ((𝐵 ∖ dom 𝐴) × {𝑋}))
188uneq2i 3797 . . . 4 (dom 𝐴 ∪ dom ((𝐵 ∖ dom 𝐴) × {𝑋})) = (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴))
1917, 18eqtri 2673 . . 3 dom (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) = (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴))
20 nodmon 31928 . . . 4 (𝐴 No → dom 𝐴 ∈ On)
21 undif 4082 . . . . . 6 (dom 𝐴𝐵 ↔ (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) = 𝐵)
22 eleq1a 2725 . . . . . . 7 (𝐵 ∈ On → ((dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) = 𝐵 → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On))
2322adantl 481 . . . . . 6 ((dom 𝐴 ∈ On ∧ 𝐵 ∈ On) → ((dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) = 𝐵 → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On))
2421, 23syl5bi 232 . . . . 5 ((dom 𝐴 ∈ On ∧ 𝐵 ∈ On) → (dom 𝐴𝐵 → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On))
25 ssdif0 3975 . . . . . 6 (𝐵 ⊆ dom 𝐴 ↔ (𝐵 ∖ dom 𝐴) = ∅)
26 uneq2 3794 . . . . . . . . . 10 ((𝐵 ∖ dom 𝐴) = ∅ → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) = (dom 𝐴 ∪ ∅))
27 un0 4000 . . . . . . . . . 10 (dom 𝐴 ∪ ∅) = dom 𝐴
2826, 27syl6eq 2701 . . . . . . . . 9 ((𝐵 ∖ dom 𝐴) = ∅ → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) = dom 𝐴)
2928eleq1d 2715 . . . . . . . 8 ((𝐵 ∖ dom 𝐴) = ∅ → ((dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On ↔ dom 𝐴 ∈ On))
3029biimprcd 240 . . . . . . 7 (dom 𝐴 ∈ On → ((𝐵 ∖ dom 𝐴) = ∅ → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On))
3130adantr 480 . . . . . 6 ((dom 𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 ∖ dom 𝐴) = ∅ → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On))
3225, 31syl5bi 232 . . . . 5 ((dom 𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ⊆ dom 𝐴 → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On))
33 eloni 5771 . . . . . 6 (dom 𝐴 ∈ On → Ord dom 𝐴)
34 eloni 5771 . . . . . 6 (𝐵 ∈ On → Ord 𝐵)
35 ordtri2or2 5861 . . . . . 6 ((Ord dom 𝐴 ∧ Ord 𝐵) → (dom 𝐴𝐵𝐵 ⊆ dom 𝐴))
3633, 34, 35syl2an 493 . . . . 5 ((dom 𝐴 ∈ On ∧ 𝐵 ∈ On) → (dom 𝐴𝐵𝐵 ⊆ dom 𝐴))
3724, 32, 36mpjaod 395 . . . 4 ((dom 𝐴 ∈ On ∧ 𝐵 ∈ On) → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On)
3820, 37sylan 487 . . 3 ((𝐴 No 𝐵 ∈ On) → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On)
3919, 38syl5eqel 2734 . 2 ((𝐴 No 𝐵 ∈ On) → dom (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) ∈ On)
40 rnun 5576 . . 3 ran (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) = (ran 𝐴 ∪ ran ((𝐵 ∖ dom 𝐴) × {𝑋}))
41 norn 31929 . . . . 5 (𝐴 No → ran 𝐴 ⊆ {1𝑜, 2𝑜})
4241adantr 480 . . . 4 ((𝐴 No 𝐵 ∈ On) → ran 𝐴 ⊆ {1𝑜, 2𝑜})
43 rnxpss 5601 . . . . 5 ran ((𝐵 ∖ dom 𝐴) × {𝑋}) ⊆ {𝑋}
44 snssi 4371 . . . . . 6 (𝑋 ∈ {1𝑜, 2𝑜} → {𝑋} ⊆ {1𝑜, 2𝑜})
452, 44ax-mp 5 . . . . 5 {𝑋} ⊆ {1𝑜, 2𝑜}
4643, 45sstri 3645 . . . 4 ran ((𝐵 ∖ dom 𝐴) × {𝑋}) ⊆ {1𝑜, 2𝑜}
47 unss 3820 . . . 4 ((ran 𝐴 ⊆ {1𝑜, 2𝑜} ∧ ran ((𝐵 ∖ dom 𝐴) × {𝑋}) ⊆ {1𝑜, 2𝑜}) ↔ (ran 𝐴 ∪ ran ((𝐵 ∖ dom 𝐴) × {𝑋})) ⊆ {1𝑜, 2𝑜})
4842, 46, 47sylanblc 697 . . 3 ((𝐴 No 𝐵 ∈ On) → (ran 𝐴 ∪ ran ((𝐵 ∖ dom 𝐴) × {𝑋})) ⊆ {1𝑜, 2𝑜})
4940, 48syl5eqss 3682 . 2 ((𝐴 No 𝐵 ∈ On) → ran (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) ⊆ {1𝑜, 2𝑜})
50 elno2 31932 . 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}))
5116, 39, 49, 50syl3anbrc 1265 1 ((𝐴 No 𝐵 ∈ On) → (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) ∈ No )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382  wa 383   = wceq 1523  wcel 2030  wne 2823  cdif 3604  cun 3605  cin 3606  wss 3607  c0 3948  {csn 4210  {cpr 4212   × cxp 5141  dom cdm 5143  ran crn 5144  Ord word 5760  Oncon0 5761  Fun wfun 5920   Fn wfn 5921  1𝑜c1o 7598  2𝑜c2o 7599   No csur 31918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-no 31921
This theorem is referenced by:  noetalem1  31988
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