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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 5542 | . 2 ⊢ ( E Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ 𝑦) = ∅)) | |
2 | vex 3499 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | zfreg 9061 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑦 ∩ 𝑥) = ∅) | |
4 | 2, 3 | mpan 688 | . . . 4 ⊢ (𝑥 ≠ ∅ → ∃𝑦 ∈ 𝑥 (𝑦 ∩ 𝑥) = ∅) |
5 | incom 4180 | . . . . . 6 ⊢ (𝑦 ∩ 𝑥) = (𝑥 ∩ 𝑦) | |
6 | 5 | eqeq1i 2828 | . . . . 5 ⊢ ((𝑦 ∩ 𝑥) = ∅ ↔ (𝑥 ∩ 𝑦) = ∅) |
7 | 6 | rexbii 3249 | . . . 4 ⊢ (∃𝑦 ∈ 𝑥 (𝑦 ∩ 𝑥) = ∅ ↔ ∃𝑦 ∈ 𝑥 (𝑥 ∩ 𝑦) = ∅) |
8 | 4, 7 | sylib 220 | . . 3 ⊢ (𝑥 ≠ ∅ → ∃𝑦 ∈ 𝑥 (𝑥 ∩ 𝑦) = ∅) |
9 | 8 | adantl 484 | . 2 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ 𝑦) = ∅) |
10 | 1, 9 | mpgbir 1800 | 1 ⊢ E Fr 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∃wrex 3141 Vcvv 3496 ∩ cin 3937 ⊆ wss 3938 ∅c0 4293 E cep 5466 Fr wfr 5513 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 ax-reg 9058 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-eprel 5467 df-fr 5516 |
This theorem is referenced by: en2lp 9071 dford2 9085 noinfep 9125 zfregs 9176 bnj852 32195 dford5reg 33029 trelpss 40794 |
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