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Mirrors > Home > MPE Home > Th. List > frc | Structured version Visualization version GIF version |
Description: Property of well-founded relation (one direction of definition using class variables). (Contributed by NM, 17-Feb-2004.) (Revised by Mario Carneiro, 19-Nov-2014.) |
Ref | Expression |
---|---|
frc.1 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
frc | ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frc.1 | . . . 4 ⊢ 𝐵 ∈ V | |
2 | fri 5591 | . . . 4 ⊢ (((𝐵 ∈ V ∧ 𝑅 Fr 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝑥) | |
3 | 1, 2 | mpanl1 698 | . . 3 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝑥) |
4 | 3 | 3impb 1115 | . 2 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝑥) |
5 | breq1 5106 | . . . 4 ⊢ (𝑦 = 𝑧 → (𝑦𝑅𝑥 ↔ 𝑧𝑅𝑥)) | |
6 | 5 | rabeq0w 4341 | . . 3 ⊢ ({𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} = ∅ ↔ ∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝑥) |
7 | 6 | rexbii 3095 | . 2 ⊢ (∃𝑥 ∈ 𝐵 {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} = ∅ ↔ ∃𝑥 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝑥) |
8 | 4, 7 | sylibr 233 | 1 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 {crab 3405 Vcvv 3443 ⊆ wss 3908 ∅c0 4280 class class class wbr 5103 Fr wfr 5583 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-br 5104 df-fr 5586 |
This theorem is referenced by: frirr 5608 epfrc 5617 |
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