![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > zfregfr | Structured version Visualization version GIF version |
Description: The membership 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 5661 | . 2 ⊢ ( E Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ 𝑦) = ∅)) | |
2 | vex 3478 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | zfreg 9589 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑦 ∩ 𝑥) = ∅) | |
4 | 2, 3 | mpan 688 | . . . 4 ⊢ (𝑥 ≠ ∅ → ∃𝑦 ∈ 𝑥 (𝑦 ∩ 𝑥) = ∅) |
5 | incom 4201 | . . . . . 6 ⊢ (𝑦 ∩ 𝑥) = (𝑥 ∩ 𝑦) | |
6 | 5 | eqeq1i 2737 | . . . . 5 ⊢ ((𝑦 ∩ 𝑥) = ∅ ↔ (𝑥 ∩ 𝑦) = ∅) |
7 | 6 | rexbii 3094 | . . . 4 ⊢ (∃𝑦 ∈ 𝑥 (𝑦 ∩ 𝑥) = ∅ ↔ ∃𝑦 ∈ 𝑥 (𝑥 ∩ 𝑦) = ∅) |
8 | 4, 7 | sylib 217 | . . 3 ⊢ (𝑥 ≠ ∅ → ∃𝑦 ∈ 𝑥 (𝑥 ∩ 𝑦) = ∅) |
9 | 8 | adantl 482 | . 2 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ 𝑦) = ∅) |
10 | 1, 9 | mpgbir 1801 | 1 ⊢ E Fr 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∃wrex 3070 Vcvv 3474 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 E cep 5579 Fr wfr 5628 |
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-10 2137 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-reg 9586 |
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-nf 1786 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-eprel 5580 df-fr 5631 |
This theorem is referenced by: en2lp 9600 dford2 9614 noinfep 9654 zfregs 9726 bnj852 33927 dford5reg 34749 trelpss 43204 |
Copyright terms: Public domain | W3C validator |