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Mathbox for Scott Fenton |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > setinds | Structured version Visualization version GIF version |
Description: Principle of set induction (or E-induction). If a property passes from all elements of 𝑥 to 𝑥 itself, then it holds for all 𝑥. (Contributed by Scott Fenton, 10-Mar-2011.) |
Ref | Expression |
---|---|
setinds.1 | ⊢ (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) |
Ref | Expression |
---|---|
setinds | ⊢ 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3452 | . 2 ⊢ 𝑥 ∈ V | |
2 | setind 9677 | . . . . 5 ⊢ (∀𝑧(𝑧 ⊆ {𝑥 ∣ 𝜑} → 𝑧 ∈ {𝑥 ∣ 𝜑}) → {𝑥 ∣ 𝜑} = V) | |
3 | dfss3 3937 | . . . . . . 7 ⊢ (𝑧 ⊆ {𝑥 ∣ 𝜑} ↔ ∀𝑦 ∈ 𝑧 𝑦 ∈ {𝑥 ∣ 𝜑}) | |
4 | df-sbc 3745 | . . . . . . . . 9 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝑦 ∈ {𝑥 ∣ 𝜑}) | |
5 | 4 | ralbii 3097 | . . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑧 [𝑦 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝑧 𝑦 ∈ {𝑥 ∣ 𝜑}) |
6 | nfcv 2908 | . . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑧 | |
7 | nfsbc1v 3764 | . . . . . . . . . . 11 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | |
8 | 6, 7 | nfralw 3297 | . . . . . . . . . 10 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝑧 [𝑦 / 𝑥]𝜑 |
9 | nfsbc1v 3764 | . . . . . . . . . 10 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑 | |
10 | 8, 9 | nfim 1900 | . . . . . . . . 9 ⊢ Ⅎ𝑥(∀𝑦 ∈ 𝑧 [𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑) |
11 | raleq 3312 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝑧 [𝑦 / 𝑥]𝜑)) | |
12 | sbceq1a 3755 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑)) | |
13 | 11, 12 | imbi12d 345 | . . . . . . . . 9 ⊢ (𝑥 = 𝑧 → ((∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) ↔ (∀𝑦 ∈ 𝑧 [𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑))) |
14 | setinds.1 | . . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) | |
15 | 10, 13, 14 | chvarfv 2234 | . . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑧 [𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑) |
16 | 5, 15 | sylbir 234 | . . . . . . 7 ⊢ (∀𝑦 ∈ 𝑧 𝑦 ∈ {𝑥 ∣ 𝜑} → [𝑧 / 𝑥]𝜑) |
17 | 3, 16 | sylbi 216 | . . . . . 6 ⊢ (𝑧 ⊆ {𝑥 ∣ 𝜑} → [𝑧 / 𝑥]𝜑) |
18 | df-sbc 3745 | . . . . . 6 ⊢ ([𝑧 / 𝑥]𝜑 ↔ 𝑧 ∈ {𝑥 ∣ 𝜑}) | |
19 | 17, 18 | sylib 217 | . . . . 5 ⊢ (𝑧 ⊆ {𝑥 ∣ 𝜑} → 𝑧 ∈ {𝑥 ∣ 𝜑}) |
20 | 2, 19 | mpg 1800 | . . . 4 ⊢ {𝑥 ∣ 𝜑} = V |
21 | 20 | eqcomi 2746 | . . 3 ⊢ V = {𝑥 ∣ 𝜑} |
22 | 21 | eqabi 2882 | . 2 ⊢ (𝑥 ∈ V ↔ 𝜑) |
23 | 1, 22 | mpbi 229 | 1 ⊢ 𝜑 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 {cab 2714 ∀wral 3065 Vcvv 3448 [wsbc 3744 ⊆ wss 3915 |
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 5247 ax-sep 5261 ax-nul 5268 ax-pr 5389 ax-un 7677 ax-reg 9535 ax-inf2 9584 |
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-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-om 7808 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 |
This theorem is referenced by: setinds2f 34393 |
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