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Mirrors > Home > MPE Home > Th. List > dfepfr | Structured version Visualization version GIF version |
Description: An alternate way of saying that the epsilon relation is well-founded. (Contributed by NM, 17-Feb-2004.) (Revised by Mario Carneiro, 23-Jun-2015.) |
Ref | Expression |
---|---|
dfepfr | ⊢ ( E Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ 𝑦) = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffr2 5411 | . 2 ⊢ ( E Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧 E 𝑦} = ∅)) | |
2 | epel 5360 | . . . . . . . 8 ⊢ (𝑧 E 𝑦 ↔ 𝑧 ∈ 𝑦) | |
3 | 2 | rabbii 3418 | . . . . . . 7 ⊢ {𝑧 ∈ 𝑥 ∣ 𝑧 E 𝑦} = {𝑧 ∈ 𝑥 ∣ 𝑧 ∈ 𝑦} |
4 | dfin5 3869 | . . . . . . 7 ⊢ (𝑥 ∩ 𝑦) = {𝑧 ∈ 𝑥 ∣ 𝑧 ∈ 𝑦} | |
5 | 3, 4 | eqtr4i 2821 | . . . . . 6 ⊢ {𝑧 ∈ 𝑥 ∣ 𝑧 E 𝑦} = (𝑥 ∩ 𝑦) |
6 | 5 | eqeq1i 2799 | . . . . 5 ⊢ ({𝑧 ∈ 𝑥 ∣ 𝑧 E 𝑦} = ∅ ↔ (𝑥 ∩ 𝑦) = ∅) |
7 | 6 | rexbii 3210 | . . . 4 ⊢ (∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧 E 𝑦} = ∅ ↔ ∃𝑦 ∈ 𝑥 (𝑥 ∩ 𝑦) = ∅) |
8 | 7 | imbi2i 337 | . . 3 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧 E 𝑦} = ∅) ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ 𝑦) = ∅)) |
9 | 8 | albii 1802 | . 2 ⊢ (∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧 E 𝑦} = ∅) ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ 𝑦) = ∅)) |
10 | 1, 9 | bitri 276 | 1 ⊢ ( E Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ 𝑦) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∀wal 1520 = wceq 1522 ≠ wne 2983 ∃wrex 3105 {crab 3108 ∩ cin 3860 ⊆ wss 3861 ∅c0 4213 class class class wbr 4964 E cep 5355 Fr wfr 5402 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-sep 5097 ax-nul 5104 ax-pr 5224 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ne 2984 df-ral 3109 df-rex 3110 df-rab 3113 df-v 3438 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-nul 4214 df-if 4384 df-sn 4475 df-pr 4477 df-op 4481 df-br 4965 df-opab 5027 df-eprel 5356 df-fr 5405 |
This theorem is referenced by: onfr 6108 zfregfr 8917 onfrALTlem3 40430 onfrALT 40435 onfrALTlem3VD 40773 onfrALTVD 40777 |
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