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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj157 | Structured version Visualization version GIF version | ||
| Description: Well-founded induction restricted to a set (𝐴 ∈ V). The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| bnj157.1 | ⊢ (𝜓 ↔ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → [𝑦 / 𝑥]𝜑)) |
| bnj157.2 | ⊢ 𝐴 ∈ V |
| bnj157.3 | ⊢ 𝑅 Fr 𝐴 |
| Ref | Expression |
|---|---|
| bnj157 | ⊢ (∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) → ∀𝑥 ∈ 𝐴 𝜑) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bnj157.3 | . 2 ⊢ 𝑅 Fr 𝐴 | |
| 2 | bnj157.2 | . . 3 ⊢ 𝐴 ∈ V | |
| 3 | bnj157.1 | . . 3 ⊢ (𝜓 ↔ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → [𝑦 / 𝑥]𝜑)) | |
| 4 | 2, 3 | bnj110 34872 | . 2 ⊢ ((𝑅 Fr 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)) → ∀𝑥 ∈ 𝐴 𝜑) |
| 5 | 1, 4 | mpan 690 | 1 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) → ∀𝑥 ∈ 𝐴 𝜑) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 [wsbc 3788 class class class wbr 5143 Fr wfr 5634 |
| 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-sep 5296 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 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-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-fr 5637 |
| This theorem is referenced by: bnj852 34935 |
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