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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 27976 | . . 3 ⊢ {〈𝑎, 𝑏〉 ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))} Fr No |
| 3 | 1 | lrrecpo 27974 | . . 3 ⊢ {〈𝑎, 𝑏〉 ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))} Po No |
| 4 | 1 | lrrecse 27975 | . . 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 27977 | . . . . 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 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 6367 | . 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 690 | 1 ⊢ (𝐴 ∈ No → 𝜒) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∪ cun 3949 {copab 5205 Po wpo 5590 Fr wfr 5634 Se wse 5635 Predcpred 6320 ‘cfv 6561 No csur 27684 L cleft 27884 R cright 27885 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-1o 8506 df-2o 8507 df-no 27687 df-slt 27688 df-bday 27689 df-sslt 27826 df-scut 27828 df-made 27886 df-old 27887 df-left 27889 df-right 27890 |
| This theorem is referenced by: addsrid 27997 negsid 28073 negsbdaylem 28088 mulsrid 28139 precsex 28242 |
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