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Mirrors > Home > MPE Home > Th. List > Mathboxes > onfrALTlem1VD | Structured version Visualization version GIF version |
Description: Virtual deduction proof of onfrALTlem1 40759.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
onfrALTlem1 40759 is onfrALTlem1VD 41101 without virtual deductions and was
automatically derived from onfrALTlem1VD 41101.
|
Ref | Expression |
---|---|
onfrALTlem1VD | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ▶ ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idn2 40824 | . . . . 5 ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ▶ (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ) | |
2 | 19.8a 2170 | . . . . 5 ⊢ ((𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) → ∃𝑥(𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)) | |
3 | 1, 2 | e2 40842 | . . . 4 ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ▶ ∃𝑥(𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ) |
4 | cbvexsv 40758 | . . . . 5 ⊢ (∃𝑥(𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ ∃𝑦[𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)) | |
5 | 4 | biimpi 217 | . . . 4 ⊢ (∃𝑥(𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) → ∃𝑦[𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)) |
6 | 3, 5 | e2 40842 | . . 3 ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ▶ ∃𝑦[𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ) |
7 | sbsbc 3773 | . . . . . 6 ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ [𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)) | |
8 | onfrALTlem4 40754 | . . . . . 6 ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)) | |
9 | 7, 8 | bitri 276 | . . . . 5 ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)) |
10 | 9 | ax-gen 1787 | . . . 4 ⊢ ∀𝑦([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)) |
11 | exbi 1838 | . . . 4 ⊢ (∀𝑦([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)) → (∃𝑦[𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ ∃𝑦(𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))) | |
12 | 10, 11 | e0a 40983 | . . 3 ⊢ (∃𝑦[𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ ∃𝑦(𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)) |
13 | 6, 12 | e2bi 40843 | . 2 ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ▶ ∃𝑦(𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅) ) |
14 | df-rex 3141 | . 2 ⊢ (∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅ ↔ ∃𝑦(𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)) | |
15 | 13, 14 | e2bir 40844 | 1 ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ▶ ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅ ) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 ∀wal 1526 = wceq 1528 ∃wex 1771 [wsb 2060 ∈ wcel 2105 ≠ wne 3013 ∃wrex 3136 [wsbc 3769 ∩ cin 3932 ⊆ wss 3933 ∅c0 4288 Oncon0 6184 ( wvd2 40788 |
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-13 2381 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-in 3940 df-nul 4289 df-vd2 40789 |
This theorem is referenced by: onfrALTVD 41102 |
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