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Mirrors > Home > MPE Home > Th. List > zfregfr | Structured version Visualization version GIF version |
Description: The epsilon relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.) |
Ref | Expression |
---|---|
zfregfr | ⊢ E Fr 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfepfr 5023 | . 2 ⊢ ( E Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ 𝑦) = ∅)) | |
2 | vex 3176 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | zfreg 8383 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑦 ∩ 𝑥) = ∅) | |
4 | 2, 3 | mpan 702 | . . . 4 ⊢ (𝑥 ≠ ∅ → ∃𝑦 ∈ 𝑥 (𝑦 ∩ 𝑥) = ∅) |
5 | incom 3767 | . . . . . 6 ⊢ (𝑦 ∩ 𝑥) = (𝑥 ∩ 𝑦) | |
6 | 5 | eqeq1i 2615 | . . . . 5 ⊢ ((𝑦 ∩ 𝑥) = ∅ ↔ (𝑥 ∩ 𝑦) = ∅) |
7 | 6 | rexbii 3023 | . . . 4 ⊢ (∃𝑦 ∈ 𝑥 (𝑦 ∩ 𝑥) = ∅ ↔ ∃𝑦 ∈ 𝑥 (𝑥 ∩ 𝑦) = ∅) |
8 | 4, 7 | sylib 207 | . . 3 ⊢ (𝑥 ≠ ∅ → ∃𝑦 ∈ 𝑥 (𝑥 ∩ 𝑦) = ∅) |
9 | 8 | adantl 481 | . 2 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ 𝑦) = ∅) |
10 | 1, 9 | mpgbir 1717 | 1 ⊢ E Fr 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∃wrex 2897 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 E cep 4947 Fr wfr 4994 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-reg 8380 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-eprel 4949 df-fr 4997 |
This theorem is referenced by: en2lp 8393 dford2 8400 noinfep 8440 zfregs 8491 bnj852 30245 dford5reg 30931 trelpss 37680 |
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