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Mirrors > Home > MPE Home > Th. List > noinds | Structured version Visualization version GIF version |
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.) |
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)) |
Ref | Expression |
---|---|
noinds | ⊢ (𝐴 ∈ No → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . . . 4 ⊢ {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))} = {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))} | |
2 | 1 | lrrecfr 27258 | . . 3 ⊢ {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))} Fr No |
3 | 1 | lrrecpo 27256 | . . 3 ⊢ {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))} Po No |
4 | 1 | lrrecse 27257 | . . 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 27259 | . . . . 5 ⊢ (𝑥 ∈ No → Pred({⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))}, No , 𝑥) = (( L ‘𝑥) ∪ ( R ‘𝑥))) |
7 | 6 | raleqdv 3314 | . . . 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 6301 | . 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) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3065 ∪ cun 3909 {copab 5168 Po wpo 5544 Fr wfr 5586 Se wse 5587 Predcpred 6253 ‘cfv 6497 No csur 26991 L cleft 27178 R cright 27179 |
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 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-1o 8413 df-2o 8414 df-no 26994 df-slt 26995 df-bday 26996 df-sslt 27124 df-scut 27126 df-made 27180 df-old 27181 df-left 27183 df-right 27184 |
This theorem is referenced by: addsid1 27279 negsid 27342 mulsid1 34414 mulsid2 34415 |
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