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Mirrors > Home > MPE Home > Th. List > oe0lem | Structured version Visualization version GIF version |
Description: A helper lemma for oe0 8136 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 413 | . . 3 ⊢ (𝜑 → (𝐴 = ∅ → 𝜓)) |
3 | 2 | adantl 482 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝜑) → (𝐴 = ∅ → 𝜓)) |
4 | on0eln0 6239 | . . . 4 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) | |
5 | 4 | adantr 481 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝜑) → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) |
6 | oe0lem.2 | . . . 4 ⊢ (((𝐴 ∈ On ∧ 𝜑) ∧ ∅ ∈ 𝐴) → 𝜓) | |
7 | 6 | ex 413 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝜑) → (∅ ∈ 𝐴 → 𝜓)) |
8 | 5, 7 | sylbird 261 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝜑) → (𝐴 ≠ ∅ → 𝜓)) |
9 | 3, 8 | pm2.61dne 3100 | 1 ⊢ ((𝐴 ∈ On ∧ 𝜑) → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∅c0 4288 Oncon0 6184 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-tr 5164 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-ord 6187 df-on 6188 |
This theorem is referenced by: oe0 8136 oev2 8137 oesuclem 8139 oecl 8151 odi 8194 oewordri 8207 oelim2 8210 oeoa 8212 oeoe 8214 |
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