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Mirrors > Home > MPE Home > Th. List > oe0lem | Structured version Visualization version GIF version |
Description: A helper lemma for oe0 8559 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 6442 | . . . 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 3026 | 1 ⊢ ((𝐴 ∈ On ∧ 𝜑) → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∅c0 4339 Oncon0 6386 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-ord 6389 df-on 6390 |
This theorem is referenced by: oe0 8559 oev2 8560 oesuclem 8562 oecl 8574 odi 8616 oewordri 8629 oelim2 8632 oeoa 8634 oeoe 8636 |
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