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Theorem noextendseq 32047
 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 32029 . . . 4 (𝐴 No → Fun 𝐴)
2 noextend.1 . . . . . 6 𝑋 ∈ {1𝑜, 2𝑜}
3 fnconstg 6206 . . . . . 6 (𝑋 ∈ {1𝑜, 2𝑜} → ((𝐵 ∖ dom 𝐴) × {𝑋}) Fn (𝐵 ∖ dom 𝐴))
4 fnfun 6101 . . . . . 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 4414 . . . . . . . . 9 (𝑋 ∈ {1𝑜, 2𝑜} → {𝑋} ≠ ∅)
7 dmxp 5451 . . . . . . . . 9 ({𝑋} ≠ ∅ → dom ((𝐵 ∖ dom 𝐴) × {𝑋}) = (𝐵 ∖ dom 𝐴))
82, 6, 7mp2b 10 . . . . . . . 8 dom ((𝐵 ∖ dom 𝐴) × {𝑋}) = (𝐵 ∖ dom 𝐴)
98ineq2i 3919 . . . . . . 7 (dom 𝐴 ∩ dom ((𝐵 ∖ dom 𝐴) × {𝑋})) = (dom 𝐴 ∩ (𝐵 ∖ dom 𝐴))
10 disjdif 4148 . . . . . . 7 (dom 𝐴 ∩ (𝐵 ∖ dom 𝐴)) = ∅
119, 10eqtri 2746 . . . . . 6 (dom 𝐴 ∩ dom ((𝐵 ∖ dom 𝐴) × {𝑋})) = ∅
12 funun 6045 . . . . . 6 (((Fun 𝐴 ∧ Fun ((𝐵 ∖ dom 𝐴) × {𝑋})) ∧ (dom 𝐴 ∩ dom ((𝐵 ∖ dom 𝐴) × {𝑋})) = ∅) → Fun (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})))
1311, 12mpan2 709 . . . . 5 ((Fun 𝐴 ∧ Fun ((𝐵 ∖ dom 𝐴) × {𝑋})) → Fun (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})))
145, 13mpan2 709 . . . 4 (Fun 𝐴 → Fun (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})))
151, 14syl 17 . . 3 (𝐴 No → Fun (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})))
1615adantr 472 . 2 ((𝐴 No 𝐵 ∈ On) → Fun (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})))
17 dmun 5438 . . . 4 dom (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) = (dom 𝐴 ∪ dom ((𝐵 ∖ dom 𝐴) × {𝑋}))
188uneq2i 3872 . . . 4 (dom 𝐴 ∪ dom ((𝐵 ∖ dom 𝐴) × {𝑋})) = (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴))
1917, 18eqtri 2746 . . 3 dom (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) = (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴))
20 nodmon 32030 . . . 4 (𝐴 No → dom 𝐴 ∈ On)
21 undif 4157 . . . . . 6 (dom 𝐴𝐵 ↔ (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) = 𝐵)
22 eleq1a 2798 . . . . . . 7 (𝐵 ∈ On → ((dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) = 𝐵 → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On))
2322adantl 473 . . . . . 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 4050 . . . . . 6 (𝐵 ⊆ dom 𝐴 ↔ (𝐵 ∖ dom 𝐴) = ∅)
26 uneq2 3869 . . . . . . . . . 10 ((𝐵 ∖ dom 𝐴) = ∅ → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) = (dom 𝐴 ∪ ∅))
27 un0 4075 . . . . . . . . . 10 (dom 𝐴 ∪ ∅) = dom 𝐴
2826, 27syl6eq 2774 . . . . . . . . 9 ((𝐵 ∖ dom 𝐴) = ∅ → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) = dom 𝐴)
2928eleq1d 2788 . . . . . . . 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 472 . . . . . 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 5846 . . . . . 6 (dom 𝐴 ∈ On → Ord dom 𝐴)
34 eloni 5846 . . . . . 6 (𝐵 ∈ On → Ord 𝐵)
35 ordtri2or2 5936 . . . . . 6 ((Ord dom 𝐴 ∧ Ord 𝐵) → (dom 𝐴𝐵𝐵 ⊆ dom 𝐴))
3633, 34, 35syl2an 495 . . . . 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 489 . . 3 ((𝐴 No 𝐵 ∈ On) → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On)
3919, 38syl5eqel 2807 . 2 ((𝐴 No 𝐵 ∈ On) → dom (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) ∈ On)
40 rnun 5651 . . 3 ran (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) = (ran 𝐴 ∪ ran ((𝐵 ∖ dom 𝐴) × {𝑋}))
41 norn 32031 . . . . 5 (𝐴 No → ran 𝐴 ⊆ {1𝑜, 2𝑜})
4241adantr 472 . . . 4 ((𝐴 No 𝐵 ∈ On) → ran 𝐴 ⊆ {1𝑜, 2𝑜})
43 rnxpss 5676 . . . . 5 ran ((𝐵 ∖ dom 𝐴) × {𝑋}) ⊆ {𝑋}
44 snssi 4447 . . . . . 6 (𝑋 ∈ {1𝑜, 2𝑜} → {𝑋} ⊆ {1𝑜, 2𝑜})
452, 44ax-mp 5 . . . . 5 {𝑋} ⊆ {1𝑜, 2𝑜}
4643, 45sstri 3718 . . . 4 ran ((𝐵 ∖ dom 𝐴) × {𝑋}) ⊆ {1𝑜, 2𝑜}
47 unss 3895 . . . 4 ((ran 𝐴 ⊆ {1𝑜, 2𝑜} ∧ ran ((𝐵 ∖ dom 𝐴) × {𝑋}) ⊆ {1𝑜, 2𝑜}) ↔ (ran 𝐴 ∪ ran ((𝐵 ∖ dom 𝐴) × {𝑋})) ⊆ {1𝑜, 2𝑜})
4842, 46, 47sylanblc 699 . . 3 ((𝐴 No 𝐵 ∈ On) → (ran 𝐴 ∪ ran ((𝐵 ∖ dom 𝐴) × {𝑋})) ⊆ {1𝑜, 2𝑜})
4940, 48syl5eqss 3755 . 2 ((𝐴 No 𝐵 ∈ On) → ran (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) ⊆ {1𝑜, 2𝑜})
50 elno2 32034 . 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 1383 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 1596   ∈ wcel 2103   ≠ wne 2896   ∖ cdif 3677   ∪ cun 3678   ∩ cin 3679   ⊆ wss 3680  ∅c0 4023  {csn 4285  {cpr 4287   × cxp 5216  dom cdm 5218  ran crn 5219  Ord word 5835  Oncon0 5836  Fun wfun 5995   Fn wfn 5996  1𝑜c1o 7673  2𝑜c2o 7674   No csur 32020 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pr 5011 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-ord 5839  df-on 5840  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-no 32023 This theorem is referenced by:  noetalem1  32090
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