Theorem noinds 28346

Description: Induction principle for a single surreal. If a property passes from a surreal's left and right sets to the surreal itself, then it holds for all surreals. (Contributed by Scott Fenton, 19-Aug-2024.)

Hypotheses

Ref	Expression
noinds.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
noinds.2	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))
noinds.3	⊢ (𝑥 ∈ No → (∀𝑦 ∈ ((L‘𝑥) ∪ (R‘𝑥))𝜓 → 𝜑))

Assertion

Ref	Expression
noinds	⊢ (𝐴 ∈ No → 𝜒)

Distinct variable groups:   𝑥,𝑦   𝑥,𝐴   𝜒,𝑥   𝜑,𝑦   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝜒(𝑦)   𝐴(𝑦)

Proof of Theorem noinds

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

Step	Hyp	Ref	Expression
1		eqid 2763	. . . 4 ⊢ {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ ((L‘𝑏) ∪ (R‘𝑏))} = {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ ((L‘𝑏) ∪ (R‘𝑏))}
2	1	lrrecfr 28344	. . 3 ⊢ {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ ((L‘𝑏) ∪ (R‘𝑏))} Fr No
3	1	lrrecpo 28342	. . 3 ⊢ {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ ((L‘𝑏) ∪ (R‘𝑏))} Po No
4	1	lrrecse 28343	. . 3 ⊢ {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ ((L‘𝑏) ∪ (R‘𝑏))} Se No
5	2, 3, 4	3pm3.2i 1358	. 2 ⊢ ({⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ ((L‘𝑏) ∪ (R‘𝑏))} Fr No ∧ {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ ((L‘𝑏) ∪ (R‘𝑏))} Po No ∧ {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ ((L‘𝑏) ∪ (R‘𝑏))} Se No)
6	1	lrrecpred 28345	. . . . 5 ⊢ (𝑥 ∈ No → Pred({⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ ((L‘𝑏) ∪ (R‘𝑏))}, No, 𝑥) = ((L‘𝑥) ∪ (R‘𝑥)))
7	6	raleqdv 3323	. . . 4 ⊢ (𝑥 ∈ No → (∀𝑦 ∈ Pred ({⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ ((L‘𝑏) ∪ (R‘𝑏))}, No, 𝑥)𝜓 ↔ ∀𝑦 ∈ ((L‘𝑥) ∪ (R‘𝑥))𝜓))
8		noinds.3	. . . 4 ⊢ (𝑥 ∈ No → (∀𝑦 ∈ ((L‘𝑥) ∪ (R‘𝑥))𝜓 → 𝜑))
9	7, 8	sylbid 243	. . 3 ⊢ (𝑥 ∈ No → (∀𝑦 ∈ Pred ({⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ ((L‘𝑏) ∪ (R‘𝑏))}, No, 𝑥)𝜓 → 𝜑))
10		noinds.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
11		noinds.2	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))
12	9, 10, 11	frpoins3g 6356	. 2 ⊢ ((({⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ ((L‘𝑏) ∪ (R‘𝑏))} Fr No ∧ {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ ((L‘𝑏) ∪ (R‘𝑏))} Po No ∧ {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ ((L‘𝑏) ∪ (R‘𝑏))} Se No) ∧ 𝐴 ∈ No) → 𝜒)
13	5, 12	mpan 703	1 ⊢ (𝐴 ∈ No → 𝜒)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103   = wceq 1569   ∈ wcel 2146  ∀wral 3080   ∪ cun 4152  {copab 5170   Po wpo 5561   Fr wfr 5605   Se wse 5606  Predcpred 6310  ‘cfv 6547  Nocsur 28012  Lcleft 28226  Rcright 28227

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7761

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  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 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6311  df-ord 6372  df-on 6373  df-suc 6375  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-2nd 8012  df-frecs 8313  df-wrecs 8344  df-recs 8393  df-1o 8490  df-2o 8491  df-no 28015  df-lts 28016  df-bday 28017  df-slts 28159  df-cuts 28161  df-made 28228  df-old 28229  df-left 28231  df-right 28232

This theorem is used by:  addsrid  28365  negsid  28442  negbdaylem  28457  mulsrid  28514  precsex  28619