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| Mirrors > Home > MPE Home > Th. List > oe0lem | Structured version Visualization version GIF version | ||
| Description: A helper lemma for oe0 8560 and others. (Contributed by NM, 6-Jan-2005.) |
| Ref | Expression |
|---|---|
| oe0lem.1 | ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝜓) |
| oe0lem.2 | ⊢ (((𝐴 ∈ On ∧ 𝜑) ∧ ∅ ∈ 𝐴) → 𝜓) |
| Ref | Expression |
|---|---|
| oe0lem | ⊢ ((𝐴 ∈ On ∧ 𝜑) → 𝜓) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | oe0lem.1 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝜓) | |
| 2 | 1 | ex 412 | . . 3 ⊢ (𝜑 → (𝐴 = ∅ → 𝜓)) |
| 3 | 2 | adantl 481 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝜑) → (𝐴 = ∅ → 𝜓)) |
| 4 | on0eln0 6440 | . . . 4 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) | |
| 5 | 4 | adantr 480 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝜑) → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) |
| 6 | oe0lem.2 | . . . 4 ⊢ (((𝐴 ∈ On ∧ 𝜑) ∧ ∅ ∈ 𝐴) → 𝜓) | |
| 7 | 6 | ex 412 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝜑) → (∅ ∈ 𝐴 → 𝜓)) |
| 8 | 5, 7 | sylbird 260 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝜑) → (𝐴 ≠ ∅ → 𝜓)) |
| 9 | 3, 8 | pm2.61dne 3028 | 1 ⊢ ((𝐴 ∈ On ∧ 𝜑) → 𝜓) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∅c0 4333 Oncon0 6384 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 |
| This theorem is referenced by: oe0 8560 oev2 8561 oesuclem 8563 oecl 8575 odi 8617 oewordri 8630 oelim2 8633 oeoa 8635 oeoe 8637 |
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