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Mirrors > Home > MPE Home > Th. List > frirr | Structured version Visualization version GIF version |
Description: A well-founded relation is irreflexive. Special case of Proposition 6.23 of [TakeutiZaring] p. 30. (Contributed by NM, 2-Jan-1994.) (Revised by Mario Carneiro, 22-Jun-2015.) |
Ref | Expression |
---|---|
frirr | ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 473 | . . 3 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝑅 Fr 𝐴) | |
2 | snssi 4339 | . . . 4 ⊢ (𝐵 ∈ 𝐴 → {𝐵} ⊆ 𝐴) | |
3 | 2 | adantl 482 | . . 3 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → {𝐵} ⊆ 𝐴) |
4 | snnzg 4308 | . . . 4 ⊢ (𝐵 ∈ 𝐴 → {𝐵} ≠ ∅) | |
5 | 4 | adantl 482 | . . 3 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → {𝐵} ≠ ∅) |
6 | snex 4908 | . . . 4 ⊢ {𝐵} ∈ V | |
7 | 6 | frc 5080 | . . 3 ⊢ ((𝑅 Fr 𝐴 ∧ {𝐵} ⊆ 𝐴 ∧ {𝐵} ≠ ∅) → ∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅) |
8 | 1, 3, 5, 7 | syl3anc 1326 | . 2 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → ∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅) |
9 | rabeq0 3957 | . . . . . 6 ⊢ ({𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ∀𝑥 ∈ {𝐵} ¬ 𝑥𝑅𝑦) | |
10 | breq2 4657 | . . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝐵)) | |
11 | 10 | notbid 308 | . . . . . . 7 ⊢ (𝑦 = 𝐵 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝐵)) |
12 | 11 | ralbidv 2986 | . . . . . 6 ⊢ (𝑦 = 𝐵 → (∀𝑥 ∈ {𝐵} ¬ 𝑥𝑅𝑦 ↔ ∀𝑥 ∈ {𝐵} ¬ 𝑥𝑅𝐵)) |
13 | 9, 12 | syl5bb 272 | . . . . 5 ⊢ (𝑦 = 𝐵 → ({𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ∀𝑥 ∈ {𝐵} ¬ 𝑥𝑅𝐵)) |
14 | 13 | rexsng 4219 | . . . 4 ⊢ (𝐵 ∈ 𝐴 → (∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ∀𝑥 ∈ {𝐵} ¬ 𝑥𝑅𝐵)) |
15 | breq1 4656 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑥𝑅𝐵 ↔ 𝐵𝑅𝐵)) | |
16 | 15 | notbid 308 | . . . . 5 ⊢ (𝑥 = 𝐵 → (¬ 𝑥𝑅𝐵 ↔ ¬ 𝐵𝑅𝐵)) |
17 | 16 | ralsng 4218 | . . . 4 ⊢ (𝐵 ∈ 𝐴 → (∀𝑥 ∈ {𝐵} ¬ 𝑥𝑅𝐵 ↔ ¬ 𝐵𝑅𝐵)) |
18 | 14, 17 | bitrd 268 | . . 3 ⊢ (𝐵 ∈ 𝐴 → (∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ¬ 𝐵𝑅𝐵)) |
19 | 18 | adantl 482 | . 2 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → (∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ¬ 𝐵𝑅𝐵)) |
20 | 8, 19 | mpbid 222 | 1 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 {crab 2916 ⊆ wss 3574 ∅c0 3915 {csn 4177 class class class wbr 4653 Fr wfr 5070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-fr 5073 |
This theorem is referenced by: efrirr 5095 predfrirr 5709 dfwe2 6981 bnj1417 31109 efrunt 31590 ifr0 38654 |
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