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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 3495 | . 2 ⊢ 𝑥 ∈ V | |
2 | setind 9164 | . . . . 5 ⊢ (∀𝑧(𝑧 ⊆ {𝑥 ∣ 𝜑} → 𝑧 ∈ {𝑥 ∣ 𝜑}) → {𝑥 ∣ 𝜑} = V) | |
3 | dfss3 3953 | . . . . . . 7 ⊢ (𝑧 ⊆ {𝑥 ∣ 𝜑} ↔ ∀𝑦 ∈ 𝑧 𝑦 ∈ {𝑥 ∣ 𝜑}) | |
4 | df-sbc 3770 | . . . . . . . . 9 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝑦 ∈ {𝑥 ∣ 𝜑}) | |
5 | 4 | ralbii 3162 | . . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑧 [𝑦 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝑧 𝑦 ∈ {𝑥 ∣ 𝜑}) |
6 | nfcv 2974 | . . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑧 | |
7 | nfsbc1v 3789 | . . . . . . . . . . 11 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | |
8 | 6, 7 | nfralw 3222 | . . . . . . . . . 10 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝑧 [𝑦 / 𝑥]𝜑 |
9 | nfsbc1v 3789 | . . . . . . . . . 10 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑 | |
10 | 8, 9 | nfim 1888 | . . . . . . . . 9 ⊢ Ⅎ𝑥(∀𝑦 ∈ 𝑧 [𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑) |
11 | raleq 3403 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝑧 [𝑦 / 𝑥]𝜑)) | |
12 | sbceq1a 3780 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑)) | |
13 | 11, 12 | imbi12d 346 | . . . . . . . . 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 2232 | . . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑧 [𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑) |
16 | 5, 15 | sylbir 236 | . . . . . . 7 ⊢ (∀𝑦 ∈ 𝑧 𝑦 ∈ {𝑥 ∣ 𝜑} → [𝑧 / 𝑥]𝜑) |
17 | 3, 16 | sylbi 218 | . . . . . 6 ⊢ (𝑧 ⊆ {𝑥 ∣ 𝜑} → [𝑧 / 𝑥]𝜑) |
18 | df-sbc 3770 | . . . . . 6 ⊢ ([𝑧 / 𝑥]𝜑 ↔ 𝑧 ∈ {𝑥 ∣ 𝜑}) | |
19 | 17, 18 | sylib 219 | . . . . 5 ⊢ (𝑧 ⊆ {𝑥 ∣ 𝜑} → 𝑧 ∈ {𝑥 ∣ 𝜑}) |
20 | 2, 19 | mpg 1789 | . . . 4 ⊢ {𝑥 ∣ 𝜑} = V |
21 | 20 | eqcomi 2827 | . . 3 ⊢ V = {𝑥 ∣ 𝜑} |
22 | 21 | abeq2i 2945 | . 2 ⊢ (𝑥 ∈ V ↔ 𝜑) |
23 | 1, 22 | mpbi 231 | 1 ⊢ 𝜑 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 {cab 2796 ∀wral 3135 Vcvv 3492 [wsbc 3769 ⊆ wss 3933 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-reg 9044 ax-inf2 9092 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 |
This theorem is referenced by: setinds2f 32921 |
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