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Mirrors > Home > MPE Home > Th. List > frd | Structured version Visualization version GIF version |
Description: A nonempty subset of an 𝑅-well-founded class has an 𝑅-minimal element (deduction form). (Contributed by BJ, 16-Nov-2024.) |
Ref | Expression |
---|---|
frd.fr | ⊢ (𝜑 → 𝑅 Fr 𝐴) |
frd.ss | ⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
frd.ex | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
frd.n0 | ⊢ (𝜑 → 𝐵 ≠ ∅) |
Ref | Expression |
---|---|
frd | ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 486 | . . 3 ⊢ ((𝜑 ∧ 𝑧 = 𝐵) → 𝑧 = 𝐵) | |
2 | biidd 262 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 = 𝐵) → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝑥)) | |
3 | 1, 2 | raleqbidv 3320 | . . 3 ⊢ ((𝜑 ∧ 𝑧 = 𝐵) → (∀𝑦 ∈ 𝑧 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)) |
4 | 1, 3 | rexeqbidv 3321 | . 2 ⊢ ((𝜑 ∧ 𝑧 = 𝐵) → (∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 ¬ 𝑦𝑅𝑥 ↔ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)) |
5 | frd.ex | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
6 | frd.ss | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) | |
7 | 5, 6 | elpwd 4567 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝐴) |
8 | frd.n0 | . . . 4 ⊢ (𝜑 → 𝐵 ≠ ∅) | |
9 | nelsn 4627 | . . . 4 ⊢ (𝐵 ≠ ∅ → ¬ 𝐵 ∈ {∅}) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (𝜑 → ¬ 𝐵 ∈ {∅}) |
11 | 7, 10 | eldifd 3922 | . 2 ⊢ (𝜑 → 𝐵 ∈ (𝒫 𝐴 ∖ {∅})) |
12 | frd.fr | . . 3 ⊢ (𝜑 → 𝑅 Fr 𝐴) | |
13 | dffr6 5592 | . . 3 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑧 ∈ (𝒫 𝐴 ∖ {∅})∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 ¬ 𝑦𝑅𝑥) | |
14 | 12, 13 | sylib 217 | . 2 ⊢ (𝜑 → ∀𝑧 ∈ (𝒫 𝐴 ∖ {∅})∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 ¬ 𝑦𝑅𝑥) |
15 | 4, 11, 14 | rspcdv2 3577 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2944 ∀wral 3065 ∃wrex 3074 ∖ cdif 3908 ⊆ wss 3911 ∅c0 4283 𝒫 cpw 4561 {csn 4587 class class class wbr 5106 Fr wfr 5586 |
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-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-ral 3066 df-rex 3075 df-v 3448 df-dif 3914 df-in 3918 df-ss 3928 df-pw 4563 df-sn 4588 df-fr 5589 |
This theorem is referenced by: fri 5594 frxp3 8084 |
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