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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 3499 | . 2 ⊢ 𝑥 ∈ V | |
2 | setind 9178 | . . . . 5 ⊢ (∀𝑧(𝑧 ⊆ {𝑥 ∣ 𝜑} → 𝑧 ∈ {𝑥 ∣ 𝜑}) → {𝑥 ∣ 𝜑} = V) | |
3 | dfss3 3958 | . . . . . . 7 ⊢ (𝑧 ⊆ {𝑥 ∣ 𝜑} ↔ ∀𝑦 ∈ 𝑧 𝑦 ∈ {𝑥 ∣ 𝜑}) | |
4 | df-sbc 3775 | . . . . . . . . 9 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝑦 ∈ {𝑥 ∣ 𝜑}) | |
5 | 4 | ralbii 3167 | . . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑧 [𝑦 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝑧 𝑦 ∈ {𝑥 ∣ 𝜑}) |
6 | nfcv 2979 | . . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑧 | |
7 | nfsbc1v 3794 | . . . . . . . . . . 11 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | |
8 | 6, 7 | nfralw 3227 | . . . . . . . . . 10 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝑧 [𝑦 / 𝑥]𝜑 |
9 | nfsbc1v 3794 | . . . . . . . . . 10 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑 | |
10 | 8, 9 | nfim 1897 | . . . . . . . . 9 ⊢ Ⅎ𝑥(∀𝑦 ∈ 𝑧 [𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑) |
11 | raleq 3407 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝑧 [𝑦 / 𝑥]𝜑)) | |
12 | sbceq1a 3785 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑)) | |
13 | 11, 12 | imbi12d 347 | . . . . . . . . 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 2242 | . . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑧 [𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑) |
16 | 5, 15 | sylbir 237 | . . . . . . 7 ⊢ (∀𝑦 ∈ 𝑧 𝑦 ∈ {𝑥 ∣ 𝜑} → [𝑧 / 𝑥]𝜑) |
17 | 3, 16 | sylbi 219 | . . . . . 6 ⊢ (𝑧 ⊆ {𝑥 ∣ 𝜑} → [𝑧 / 𝑥]𝜑) |
18 | df-sbc 3775 | . . . . . 6 ⊢ ([𝑧 / 𝑥]𝜑 ↔ 𝑧 ∈ {𝑥 ∣ 𝜑}) | |
19 | 17, 18 | sylib 220 | . . . . 5 ⊢ (𝑧 ⊆ {𝑥 ∣ 𝜑} → 𝑧 ∈ {𝑥 ∣ 𝜑}) |
20 | 2, 19 | mpg 1798 | . . . 4 ⊢ {𝑥 ∣ 𝜑} = V |
21 | 20 | eqcomi 2832 | . . 3 ⊢ V = {𝑥 ∣ 𝜑} |
22 | 21 | abeq2i 2950 | . 2 ⊢ (𝑥 ∈ V ↔ 𝜑) |
23 | 1, 22 | mpbi 232 | 1 ⊢ 𝜑 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 {cab 2801 ∀wral 3140 Vcvv 3496 [wsbc 3774 ⊆ wss 3938 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-reg 9058 ax-inf2 9106 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 |
This theorem is referenced by: setinds2f 33026 |
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