Theorem noinds 27996

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 2740	. . . 4 ⊢ {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))} = {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))}
2	1	lrrecfr 27994	. . 3 ⊢ {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))} Fr No
3	1	lrrecpo 27992	. . 3 ⊢ {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))} Po No
4	1	lrrecse 27993	. . 3 ⊢ {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))} Se No
5	2, 3, 4	3pm3.2i 1339	. 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 27995	. . . . 5 ⊢ (𝑥 ∈ No → Pred({⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))}, No , 𝑥) = (( L ‘𝑥) ∪ ( R ‘𝑥)))
7	6	raleqdv 3334	. . . 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 240	. . 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 6378	. 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 689	1 ⊢ (𝐴 ∈ No → 𝜒)

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067   ∪ cun 3974  {copab 5228   Po wpo 5605   Fr wfr 5649   Se wse 5650  Predcpred 6331  ‘cfv 6573   No csur 27702   L cleft 27902   R cright 27903

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-1o 8522  df-2o 8523  df-no 27705  df-slt 27706  df-bday 27707  df-sslt 27844  df-scut 27846  df-made 27904  df-old 27905  df-left 27907  df-right 27908

This theorem is referenced by:  addsrid  28015  negsid  28091  negsbdaylem  28106  mulsrid  28157  precsex  28260