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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 5622 | . 2 ⊢ ( E Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ 𝑦) = ∅)) | |
2 | vex 3451 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | zfreg 9539 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑦 ∩ 𝑥) = ∅) | |
4 | 2, 3 | mpan 689 | . . . 4 ⊢ (𝑥 ≠ ∅ → ∃𝑦 ∈ 𝑥 (𝑦 ∩ 𝑥) = ∅) |
5 | incom 4165 | . . . . . 6 ⊢ (𝑦 ∩ 𝑥) = (𝑥 ∩ 𝑦) | |
6 | 5 | eqeq1i 2738 | . . . . 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 483 | . 2 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ 𝑦) = ∅) |
10 | 1, 9 | mpgbir 1802 | 1 ⊢ E Fr 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 ∃wrex 3070 Vcvv 3447 ∩ cin 3913 ⊆ wss 3914 ∅c0 4286 E cep 5540 Fr wfr 5589 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 ax-reg 9536 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-br 5110 df-opab 5172 df-eprel 5541 df-fr 5592 |
This theorem is referenced by: en2lp 9550 dford2 9564 noinfep 9604 zfregs 9676 bnj852 33597 dford5reg 34420 trelpss 42827 |
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