Theorem noinds 27419

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 2733	. . . 4 ⊢ {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))} = {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))}
2	1	lrrecfr 27417	. . 3 ⊢ {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))} Fr No
3	1	lrrecpo 27415	. . 3 ⊢ {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))} Po No
4	1	lrrecse 27416	. . 3 ⊢ {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))} Se No
5	2, 3, 4	3pm3.2i 1340	. 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 27418	. . . . 5 ⊢ (𝑥 ∈ No → Pred({⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))}, No , 𝑥) = (( L ‘𝑥) ∪ ( R ‘𝑥)))
7	6	raleqdv 3326	. . . 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 239	. . 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 6345	. 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 205   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ∀wral 3062   ∪ cun 3946  {copab 5210   Po wpo 5586   Fr wfr 5628   Se wse 5629  Predcpred 6297  ‘cfv 6541   No csur 27133   L cleft 27330   R cright 27331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-1o 8463  df-2o 8464  df-no 27136  df-slt 27137  df-bday 27138  df-sslt 27273  df-scut 27275  df-made 27332  df-old 27333  df-left 27335  df-right 27336

This theorem is referenced by:  addsrid  27438  negsid  27505  negsbdaylem  27520  mulsrid  27559  precsex  27654