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Theorem noseqinds 28314
Description: Induction schema for surreal sequences. (Contributed by Scott Fenton, 18-Apr-2025.)
Hypotheses
Ref Expression
noseq.1 (𝜑𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω))
noseq.2 (𝜑𝐴 No )
noseqinds.3 (𝑦 = 𝐴 → (𝜓𝜒))
noseqinds.4 (𝑦 = 𝑧 → (𝜓𝜃))
noseqinds.5 (𝑦 = (𝑧 +s 1s ) → (𝜓𝜏))
noseqinds.6 (𝑦 = 𝐵 → (𝜓𝜂))
noseqinds.7 (𝜑𝜒)
noseqinds.8 ((𝜑𝑧𝑍) → (𝜃𝜏))
Assertion
Ref Expression
noseqinds ((𝜑𝐵𝑍) → 𝜂)
Distinct variable groups:   𝑦,𝑧,𝐴   𝑦,𝐵   𝜒,𝑦   𝜂,𝑦   𝜑,𝑧   𝜓,𝑧   𝜏,𝑦   𝜃,𝑦   𝑦,𝑍,𝑧   𝑥,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑧)   𝜃(𝑥,𝑧)   𝜏(𝑥,𝑧)   𝜂(𝑥,𝑧)   𝐴(𝑥)   𝐵(𝑥,𝑧)   𝑍(𝑥)

Proof of Theorem noseqinds
StepHypRef Expression
1 noseq.1 . . . . 5 (𝜑𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω))
2 noseq.2 . . . . 5 (𝜑𝐴 No )
3 noseqinds.3 . . . . . 6 (𝑦 = 𝐴 → (𝜓𝜒))
41, 2noseq0 28311 . . . . . 6 (𝜑𝐴𝑍)
5 noseqinds.7 . . . . . 6 (𝜑𝜒)
63, 4, 5elrabd 3697 . . . . 5 (𝜑𝐴 ∈ {𝑦𝑍𝜓})
7 noseqinds.8 . . . . . . . . 9 ((𝜑𝑧𝑍) → (𝜃𝜏))
81adantr 480 . . . . . . . . . 10 ((𝜑𝑧𝑍) → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω))
92adantr 480 . . . . . . . . . 10 ((𝜑𝑧𝑍) → 𝐴 No )
10 simpr 484 . . . . . . . . . 10 ((𝜑𝑧𝑍) → 𝑧𝑍)
118, 9, 10noseqp1 28312 . . . . . . . . 9 ((𝜑𝑧𝑍) → (𝑧 +s 1s ) ∈ 𝑍)
127, 11jctild 525 . . . . . . . 8 ((𝜑𝑧𝑍) → (𝜃 → ((𝑧 +s 1s ) ∈ 𝑍𝜏)))
1312expimpd 453 . . . . . . 7 (𝜑 → ((𝑧𝑍𝜃) → ((𝑧 +s 1s ) ∈ 𝑍𝜏)))
14 noseqinds.4 . . . . . . . 8 (𝑦 = 𝑧 → (𝜓𝜃))
1514elrab 3695 . . . . . . 7 (𝑧 ∈ {𝑦𝑍𝜓} ↔ (𝑧𝑍𝜃))
16 noseqinds.5 . . . . . . . 8 (𝑦 = (𝑧 +s 1s ) → (𝜓𝜏))
1716elrab 3695 . . . . . . 7 ((𝑧 +s 1s ) ∈ {𝑦𝑍𝜓} ↔ ((𝑧 +s 1s ) ∈ 𝑍𝜏))
1813, 15, 173imtr4g 296 . . . . . 6 (𝜑 → (𝑧 ∈ {𝑦𝑍𝜓} → (𝑧 +s 1s ) ∈ {𝑦𝑍𝜓}))
1918imp 406 . . . . 5 ((𝜑𝑧 ∈ {𝑦𝑍𝜓}) → (𝑧 +s 1s ) ∈ {𝑦𝑍𝜓})
201, 2, 6, 19noseqind 28313 . . . 4 (𝜑𝑍 ⊆ {𝑦𝑍𝜓})
2120sselda 3995 . . 3 ((𝜑𝐵𝑍) → 𝐵 ∈ {𝑦𝑍𝜓})
22 noseqinds.6 . . . 4 (𝑦 = 𝐵 → (𝜓𝜂))
2322elrab 3695 . . 3 (𝐵 ∈ {𝑦𝑍𝜓} ↔ (𝐵𝑍𝜂))
2421, 23sylib 218 . 2 ((𝜑𝐵𝑍) → (𝐵𝑍𝜂))
2524simprd 495 1 ((𝜑𝐵𝑍) → 𝜂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  {crab 3433  Vcvv 3478  cmpt 5231  cima 5692  (class class class)co 7431  ωcom 7887  reccrdg 8448   No csur 27699   1s c1s 27883   +s cadds 28007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449
This theorem is referenced by:  n0sind  28352  nnsind  28378
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