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Mirrors > Home > MPE Home > Th. List > r19.2zb | Structured version Visualization version GIF version |
Description: A response to the notion that the condition 𝐴 ≠ ∅ can be removed in r19.2z 4283. Interestingly enough, 𝜑 does not figure in the left-hand side. (Contributed by Jeff Hankins, 24-Aug-2009.) |
Ref | Expression |
---|---|
r19.2zb | ⊢ (𝐴 ≠ ∅ ↔ (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.2z 4283 | . . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝜑) → ∃𝑥 ∈ 𝐴 𝜑) | |
2 | 1 | ex 403 | . 2 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜑)) |
3 | noel 4149 | . . . . . . 7 ⊢ ¬ 𝑥 ∈ ∅ | |
4 | 3 | pm2.21i 117 | . . . . . 6 ⊢ (𝑥 ∈ ∅ → 𝜑) |
5 | 4 | rgen 3132 | . . . . 5 ⊢ ∀𝑥 ∈ ∅ 𝜑 |
6 | raleq 3351 | . . . . 5 ⊢ (𝐴 = ∅ → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ ∅ 𝜑)) | |
7 | 5, 6 | mpbiri 250 | . . . 4 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑) |
8 | 7 | necon3bi 3026 | . . 3 ⊢ (¬ ∀𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅) |
9 | exsimpl 1971 | . . . 4 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥 𝑥 ∈ 𝐴) | |
10 | df-rex 3124 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
11 | n0 4161 | . . . 4 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
12 | 9, 10, 11 | 3imtr4i 284 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅) |
13 | 8, 12 | ja 175 | . 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜑) → 𝐴 ≠ ∅) |
14 | 2, 13 | impbii 201 | 1 ⊢ (𝐴 ≠ ∅ ↔ (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1658 ∃wex 1880 ∈ wcel 2166 ≠ wne 3000 ∀wral 3118 ∃wrex 3119 ∅c0 4145 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-ral 3123 df-rex 3124 df-v 3417 df-dif 3802 df-nul 4146 |
This theorem is referenced by: iinpreima 6595 utopbas 22410 clsk3nimkb 39179 radcnvrat 39354 |
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