Theorem noinds 28331

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 2761	. . . 4 ⊢ {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ ((L‘𝑏) ∪ (R‘𝑏))} = {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ ((L‘𝑏) ∪ (R‘𝑏))}
2	1	lrrecfr 28329	. . 3 ⊢ {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ ((L‘𝑏) ∪ (R‘𝑏))} Fr No
3	1	lrrecpo 28327	. . 3 ⊢ {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ ((L‘𝑏) ∪ (R‘𝑏))} Po No
4	1	lrrecse 28328	. . 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 28330	. . . . 5 ⊢ (𝑥 ∈ No → Pred({⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ ((L‘𝑏) ∪ (R‘𝑏))}, No, 𝑥) = ((L‘𝑥) ∪ (R‘𝑥)))
7	6	raleqdv 3320	. . . 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 6349	. 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 1570   ∈ wcel 2145  ∀wral 3077   ∪ cun 3897  {copab 5167   Po wpo 5557   Fr wfr 5601   Se wse 5602  Predcpred 6303  ‘cfv 6538  Nocsur 27997  Lcleft 28211  Rcright 28212

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-1o 8476  df-2o 8477  df-no 28000  df-lts 28001  df-bday 28002  df-slts 28144  df-cuts 28146  df-made 28213  df-old 28214  df-left 28216  df-right 28217

This theorem is used by:  addsrid  28350  negsid  28427  negbdaylem  28442  mulsrid  28499  precsex  28604