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Mirrors > Home > MPE Home > Th. List > oe0lem | Structured version Visualization version GIF version |
Description: A helper lemma for oe0 8130 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 416 | . . 3 ⊢ (𝜑 → (𝐴 = ∅ → 𝜓)) |
3 | 2 | adantl 485 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝜑) → (𝐴 = ∅ → 𝜓)) |
4 | on0eln0 6214 | . . . 4 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) | |
5 | 4 | adantr 484 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝜑) → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) |
6 | oe0lem.2 | . . . 4 ⊢ (((𝐴 ∈ On ∧ 𝜑) ∧ ∅ ∈ 𝐴) → 𝜓) | |
7 | 6 | ex 416 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝜑) → (∅ ∈ 𝐴 → 𝜓)) |
8 | 5, 7 | sylbird 263 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝜑) → (𝐴 ≠ ∅ → 𝜓)) |
9 | 3, 8 | pm2.61dne 3073 | 1 ⊢ ((𝐴 ∈ On ∧ 𝜑) → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∅c0 4243 Oncon0 6159 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-tr 5137 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-ord 6162 df-on 6163 |
This theorem is referenced by: oe0 8130 oev2 8131 oesuclem 8133 oecl 8145 odi 8188 oewordri 8201 oelim2 8204 oeoa 8206 oeoe 8208 |
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