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Theorem noextendseq 33302
 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 33284 . . . 4 (𝐴 No → Fun 𝐴)
2 noextend.1 . . . . 5 𝑋 ∈ {1o, 2o}
3 fnconstg 6542 . . . . 5 (𝑋 ∈ {1o, 2o} → ((𝐵 ∖ dom 𝐴) × {𝑋}) Fn (𝐵 ∖ dom 𝐴))
4 fnfun 6424 . . . . 5 (((𝐵 ∖ dom 𝐴) × {𝑋}) Fn (𝐵 ∖ dom 𝐴) → Fun ((𝐵 ∖ dom 𝐴) × {𝑋}))
52, 3, 4mp2b 10 . . . 4 Fun ((𝐵 ∖ dom 𝐴) × {𝑋})
6 snnzg 4670 . . . . . . . 8 (𝑋 ∈ {1o, 2o} → {𝑋} ≠ ∅)
7 dmxp 5764 . . . . . . . 8 ({𝑋} ≠ ∅ → dom ((𝐵 ∖ dom 𝐴) × {𝑋}) = (𝐵 ∖ dom 𝐴))
82, 6, 7mp2b 10 . . . . . . 7 dom ((𝐵 ∖ dom 𝐴) × {𝑋}) = (𝐵 ∖ dom 𝐴)
98ineq2i 4136 . . . . . 6 (dom 𝐴 ∩ dom ((𝐵 ∖ dom 𝐴) × {𝑋})) = (dom 𝐴 ∩ (𝐵 ∖ dom 𝐴))
10 disjdif 4379 . . . . . 6 (dom 𝐴 ∩ (𝐵 ∖ dom 𝐴)) = ∅
119, 10eqtri 2821 . . . . 5 (dom 𝐴 ∩ dom ((𝐵 ∖ dom 𝐴) × {𝑋})) = ∅
12 funun 6371 . . . . 5 (((Fun 𝐴 ∧ Fun ((𝐵 ∖ dom 𝐴) × {𝑋})) ∧ (dom 𝐴 ∩ dom ((𝐵 ∖ dom 𝐴) × {𝑋})) = ∅) → Fun (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})))
1311, 12mpan2 690 . . . 4 ((Fun 𝐴 ∧ Fun ((𝐵 ∖ dom 𝐴) × {𝑋})) → Fun (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})))
141, 5, 13sylancl 589 . . 3 (𝐴 No → Fun (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})))
1514adantr 484 . 2 ((𝐴 No 𝐵 ∈ On) → Fun (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})))
16 dmun 5744 . . . 4 dom (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) = (dom 𝐴 ∪ dom ((𝐵 ∖ dom 𝐴) × {𝑋}))
178uneq2i 4087 . . . 4 (dom 𝐴 ∪ dom ((𝐵 ∖ dom 𝐴) × {𝑋})) = (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴))
1816, 17eqtri 2821 . . 3 dom (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) = (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴))
19 nodmon 33285 . . . 4 (𝐴 No → dom 𝐴 ∈ On)
20 undif 4388 . . . . . 6 (dom 𝐴𝐵 ↔ (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) = 𝐵)
21 eleq1a 2885 . . . . . . 7 (𝐵 ∈ On → ((dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) = 𝐵 → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On))
2221adantl 485 . . . . . 6 ((dom 𝐴 ∈ On ∧ 𝐵 ∈ On) → ((dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) = 𝐵 → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On))
2320, 22syl5bi 245 . . . . 5 ((dom 𝐴 ∈ On ∧ 𝐵 ∈ On) → (dom 𝐴𝐵 → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On))
24 ssdif0 4277 . . . . . 6 (𝐵 ⊆ dom 𝐴 ↔ (𝐵 ∖ dom 𝐴) = ∅)
25 uneq2 4084 . . . . . . . . . 10 ((𝐵 ∖ dom 𝐴) = ∅ → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) = (dom 𝐴 ∪ ∅))
26 un0 4298 . . . . . . . . . 10 (dom 𝐴 ∪ ∅) = dom 𝐴
2725, 26eqtrdi 2849 . . . . . . . . 9 ((𝐵 ∖ dom 𝐴) = ∅ → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) = dom 𝐴)
2827eleq1d 2874 . . . . . . . 8 ((𝐵 ∖ dom 𝐴) = ∅ → ((dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On ↔ dom 𝐴 ∈ On))
2928biimprcd 253 . . . . . . 7 (dom 𝐴 ∈ On → ((𝐵 ∖ dom 𝐴) = ∅ → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On))
3029adantr 484 . . . . . 6 ((dom 𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 ∖ dom 𝐴) = ∅ → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On))
3124, 30syl5bi 245 . . . . 5 ((dom 𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ⊆ dom 𝐴 → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On))
32 eloni 6170 . . . . . 6 (dom 𝐴 ∈ On → Ord dom 𝐴)
33 eloni 6170 . . . . . 6 (𝐵 ∈ On → Ord 𝐵)
34 ordtri2or2 6256 . . . . . 6 ((Ord dom 𝐴 ∧ Ord 𝐵) → (dom 𝐴𝐵𝐵 ⊆ dom 𝐴))
3532, 33, 34syl2an 598 . . . . 5 ((dom 𝐴 ∈ On ∧ 𝐵 ∈ On) → (dom 𝐴𝐵𝐵 ⊆ dom 𝐴))
3623, 31, 35mpjaod 857 . . . 4 ((dom 𝐴 ∈ On ∧ 𝐵 ∈ On) → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On)
3719, 36sylan 583 . . 3 ((𝐴 No 𝐵 ∈ On) → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On)
3818, 37eqeltrid 2894 . 2 ((𝐴 No 𝐵 ∈ On) → dom (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) ∈ On)
39 rnun 5972 . . 3 ran (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) = (ran 𝐴 ∪ ran ((𝐵 ∖ dom 𝐴) × {𝑋}))
40 norn 33286 . . . . 5 (𝐴 No → ran 𝐴 ⊆ {1o, 2o})
4140adantr 484 . . . 4 ((𝐴 No 𝐵 ∈ On) → ran 𝐴 ⊆ {1o, 2o})
42 rnxpss 5997 . . . . 5 ran ((𝐵 ∖ dom 𝐴) × {𝑋}) ⊆ {𝑋}
43 snssi 4701 . . . . . 6 (𝑋 ∈ {1o, 2o} → {𝑋} ⊆ {1o, 2o})
442, 43ax-mp 5 . . . . 5 {𝑋} ⊆ {1o, 2o}
4542, 44sstri 3924 . . . 4 ran ((𝐵 ∖ dom 𝐴) × {𝑋}) ⊆ {1o, 2o}
46 unss 4111 . . . 4 ((ran 𝐴 ⊆ {1o, 2o} ∧ ran ((𝐵 ∖ dom 𝐴) × {𝑋}) ⊆ {1o, 2o}) ↔ (ran 𝐴 ∪ ran ((𝐵 ∖ dom 𝐴) × {𝑋})) ⊆ {1o, 2o})
4741, 45, 46sylanblc 592 . . 3 ((𝐴 No 𝐵 ∈ On) → (ran 𝐴 ∪ ran ((𝐵 ∖ dom 𝐴) × {𝑋})) ⊆ {1o, 2o})
4839, 47eqsstrid 3963 . 2 ((𝐴 No 𝐵 ∈ On) → ran (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) ⊆ {1o, 2o})
49 elno2 33289 . 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}))
5015, 38, 48, 49syl3anbrc 1340 1 ((𝐴 No 𝐵 ∈ On) → (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) ∈ No )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111   ≠ wne 2987   ∖ cdif 3878   ∪ cun 3879   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243  {csn 4525  {cpr 4527   × cxp 5518  dom cdm 5520  ran crn 5521  Ord word 6159  Oncon0 6160  Fun wfun 6319   Fn wfn 6320  1oc1o 8081  2oc2o 8082   No csur 33275 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-ord 6163  df-on 6164  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-no 33278 This theorem is referenced by:  noetalem1  33345
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