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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 5510 | . . . 4 ⊢ (((𝐵 ∈ V ∧ 𝑅 Fr 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) | |
3 | 1, 2 | mpanl1 698 | . . 3 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) |
4 | 3 | 3impb 1110 | . 2 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) |
5 | rabeq0 4336 | . . 3 ⊢ ({𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} = ∅ ↔ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) | |
6 | 5 | rexbii 3245 | . 2 ⊢ (∃𝑥 ∈ 𝐵 {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} = ∅ ↔ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) |
7 | 4, 6 | sylibr 236 | 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 398 ∧ w3a 1082 = wceq 1531 ∈ wcel 2108 ≠ wne 3014 ∀wral 3136 ∃wrex 3137 {crab 3140 Vcvv 3493 ⊆ wss 3934 ∅c0 4289 class class class wbr 5057 Fr wfr 5504 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-dif 3937 df-in 3941 df-ss 3950 df-nul 4290 df-fr 5507 |
This theorem is referenced by: frirr 5525 epfrc 5534 |
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