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Mirrors > Home > MPE Home > Th. List > oe0lem | Structured version Visualization version GIF version |
Description: A helper lemma for oe0 7771 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 449 | . . 3 ⊢ (𝜑 → (𝐴 = ∅ → 𝜓)) |
3 | 2 | adantl 473 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝜑) → (𝐴 = ∅ → 𝜓)) |
4 | on0eln0 5941 | . . . 4 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) | |
5 | 4 | adantr 472 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝜑) → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) |
6 | oe0lem.2 | . . . 4 ⊢ (((𝐴 ∈ On ∧ 𝜑) ∧ ∅ ∈ 𝐴) → 𝜓) | |
7 | 6 | ex 449 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝜑) → (∅ ∈ 𝐴 → 𝜓)) |
8 | 5, 7 | sylbird 250 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝜑) → (𝐴 ≠ ∅ → 𝜓)) |
9 | 3, 8 | pm2.61dne 3018 | 1 ⊢ ((𝐴 ∈ On ∧ 𝜑) → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ≠ wne 2932 ∅c0 4058 Oncon0 5884 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pr 5055 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-tr 4905 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-ord 5887 df-on 5888 |
This theorem is referenced by: oe0 7771 oev2 7772 oesuclem 7774 oecl 7786 odi 7828 oewordri 7841 oelim2 7844 oeoa 7846 oeoe 7848 |
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