Theorem no2inds 28398

Description: Double induction on surreals. The many substitution instances are to cover all possible cases. (Contributed by Scott Fenton, 22-Aug-2024.)

Hypotheses

Ref	Expression
no2inds.1	⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓))
no2inds.2	⊢ (𝑦 = 𝑤 → (𝜓 ↔ 𝜒))
no2inds.3	⊢ (𝑥 = 𝑧 → (𝜃 ↔ 𝜒))
no2inds.4	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
no2inds.5	⊢ (𝑦 = 𝐵 → (𝜏 ↔ 𝜂))
no2inds.i	⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No) → ((∀𝑧 ∈ ((L‘𝑥) ∪ (R‘𝑥))∀𝑤 ∈ ((L‘𝑦) ∪ (R‘𝑦))𝜒 ∧ ∀𝑧 ∈ ((L‘𝑥) ∪ (R‘𝑥))𝜓 ∧ ∀𝑤 ∈ ((L‘𝑦) ∪ (R‘𝑦))𝜃) → 𝜑))

Assertion

Ref	Expression
no2inds	⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → 𝜂)

Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐵   𝜒,𝑦   𝜂,𝑦   𝜑,𝑧   𝜓,𝑤,𝑥   𝜏,𝑥   𝜃,𝑧   𝑥,𝑤,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑤)   𝜓(𝑦, 𝑧)   𝜒(𝑥, 𝑧, 𝑤)   𝜃(𝑥, 𝑦, 𝑤)   𝜏(𝑦, 𝑧, 𝑤)   𝜂(𝑥, 𝑧, 𝑤)   𝐴(𝑧, 𝑤)   𝐵(𝑥, 𝑧, 𝑤)

Proof of Theorem no2inds

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2763	. 2 ⊢ {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ ((L‘𝑏) ∪ (R‘𝑏))} = {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ ((L‘𝑏) ∪ (R‘𝑏))}
2		no2inds.1	. 2 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓))
3		no2inds.2	. 2 ⊢ (𝑦 = 𝑤 → (𝜓 ↔ 𝜒))
4		no2inds.3	. 2 ⊢ (𝑥 = 𝑧 → (𝜃 ↔ 𝜒))
5		no2inds.4	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
6		no2inds.5	. 2 ⊢ (𝑦 = 𝐵 → (𝜏 ↔ 𝜂))
7		no2inds.i	. 2 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No) → ((∀𝑧 ∈ ((L‘𝑥) ∪ (R‘𝑥))∀𝑤 ∈ ((L‘𝑦) ∪ (R‘𝑦))𝜒 ∧ ∀𝑧 ∈ ((L‘𝑥) ∪ (R‘𝑥))𝜓 ∧ ∀𝑤 ∈ ((L‘𝑦) ∪ (R‘𝑦))𝜃) → 𝜑))
8	1, 2, 3, 4, 5, 6, 7	no2indlesm 28397	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → 𝜂)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080   ∪ cun 4152  {copab 5173  ‘cfv 6566  Nocsur 28054  Lcleft 28268  Rcright 28269

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-1o 8514  df-2o 8515  df-no 28057  df-lts 28058  df-bday 28059  df-slts 28201  df-cuts 28203  df-made 28270  df-old 28271  df-left 28273  df-right 28274

This theorem is used by:  addscom  28409  addbday  28461  mulscom  28582