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Mirrors > Home > MPE Home > Th. List > noseqinds | Structured version Visualization version GIF version |
Description: Induction schema for surreal sequences. (Contributed by Scott Fenton, 18-Apr-2025.) |
Ref | Expression |
---|---|
noseq.1 | ⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω)) |
noseq.2 | ⊢ (𝜑 → 𝐴 ∈ No ) |
noseqinds.3 | ⊢ (𝑦 = 𝐴 → (𝜓 ↔ 𝜒)) |
noseqinds.4 | ⊢ (𝑦 = 𝑧 → (𝜓 ↔ 𝜃)) |
noseqinds.5 | ⊢ (𝑦 = (𝑧 +s 1s ) → (𝜓 ↔ 𝜏)) |
noseqinds.6 | ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜂)) |
noseqinds.7 | ⊢ (𝜑 → 𝜒) |
noseqinds.8 | ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑍) → (𝜃 → 𝜏)) |
Ref | Expression |
---|---|
noseqinds | ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → 𝜂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noseq.1 | . . . . 5 ⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω)) | |
2 | noseq.2 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ No ) | |
3 | noseqinds.3 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (𝜓 ↔ 𝜒)) | |
4 | 1, 2 | noseq0 28311 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑍) |
5 | noseqinds.7 | . . . . . 6 ⊢ (𝜑 → 𝜒) | |
6 | 3, 4, 5 | elrabd 3697 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ {𝑦 ∈ 𝑍 ∣ 𝜓}) |
7 | noseqinds.8 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑍) → (𝜃 → 𝜏)) | |
8 | 1 | adantr 480 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑍) → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω)) |
9 | 2 | adantr 480 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑍) → 𝐴 ∈ No ) |
10 | simpr 484 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑍) → 𝑧 ∈ 𝑍) | |
11 | 8, 9, 10 | noseqp1 28312 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑍) → (𝑧 +s 1s ) ∈ 𝑍) |
12 | 7, 11 | jctild 525 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑍) → (𝜃 → ((𝑧 +s 1s ) ∈ 𝑍 ∧ 𝜏))) |
13 | 12 | expimpd 453 | . . . . . . 7 ⊢ (𝜑 → ((𝑧 ∈ 𝑍 ∧ 𝜃) → ((𝑧 +s 1s ) ∈ 𝑍 ∧ 𝜏))) |
14 | noseqinds.4 | . . . . . . . 8 ⊢ (𝑦 = 𝑧 → (𝜓 ↔ 𝜃)) | |
15 | 14 | elrab 3695 | . . . . . . 7 ⊢ (𝑧 ∈ {𝑦 ∈ 𝑍 ∣ 𝜓} ↔ (𝑧 ∈ 𝑍 ∧ 𝜃)) |
16 | noseqinds.5 | . . . . . . . 8 ⊢ (𝑦 = (𝑧 +s 1s ) → (𝜓 ↔ 𝜏)) | |
17 | 16 | elrab 3695 | . . . . . . 7 ⊢ ((𝑧 +s 1s ) ∈ {𝑦 ∈ 𝑍 ∣ 𝜓} ↔ ((𝑧 +s 1s ) ∈ 𝑍 ∧ 𝜏)) |
18 | 13, 15, 17 | 3imtr4g 296 | . . . . . 6 ⊢ (𝜑 → (𝑧 ∈ {𝑦 ∈ 𝑍 ∣ 𝜓} → (𝑧 +s 1s ) ∈ {𝑦 ∈ 𝑍 ∣ 𝜓})) |
19 | 18 | imp 406 | . . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ {𝑦 ∈ 𝑍 ∣ 𝜓}) → (𝑧 +s 1s ) ∈ {𝑦 ∈ 𝑍 ∣ 𝜓}) |
20 | 1, 2, 6, 19 | noseqind 28313 | . . . 4 ⊢ (𝜑 → 𝑍 ⊆ {𝑦 ∈ 𝑍 ∣ 𝜓}) |
21 | 20 | sselda 3995 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → 𝐵 ∈ {𝑦 ∈ 𝑍 ∣ 𝜓}) |
22 | noseqinds.6 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜂)) | |
23 | 22 | elrab 3695 | . . 3 ⊢ (𝐵 ∈ {𝑦 ∈ 𝑍 ∣ 𝜓} ↔ (𝐵 ∈ 𝑍 ∧ 𝜂)) |
24 | 21, 23 | sylib 218 | . 2 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (𝐵 ∈ 𝑍 ∧ 𝜂)) |
25 | 24 | simprd 495 | 1 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → 𝜂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {crab 3433 Vcvv 3478 ↦ cmpt 5231 “ cima 5692 (class class class)co 7431 ωcom 7887 reccrdg 8448 No csur 27699 1s c1s 27883 +s cadds 28007 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 |
This theorem is referenced by: n0sind 28352 nnsind 28378 |
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