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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 4501. 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 4501 | . . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝜑) → ∃𝑥 ∈ 𝐴 𝜑) | |
2 | 1 | ex 412 | . 2 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜑)) |
3 | noel 4344 | . . . . . . 7 ⊢ ¬ 𝑥 ∈ ∅ | |
4 | 3 | pm2.21i 119 | . . . . . 6 ⊢ (𝑥 ∈ ∅ → 𝜑) |
5 | 4 | rgen 3061 | . . . . 5 ⊢ ∀𝑥 ∈ ∅ 𝜑 |
6 | raleq 3321 | . . . . 5 ⊢ (𝐴 = ∅ → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ ∅ 𝜑)) | |
7 | 5, 6 | mpbiri 258 | . . . 4 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑) |
8 | 7 | necon3bi 2965 | . . 3 ⊢ (¬ ∀𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅) |
9 | exsimpl 1866 | . . . 4 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥 𝑥 ∈ 𝐴) | |
10 | df-rex 3069 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
11 | n0 4359 | . . . 4 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
12 | 9, 10, 11 | 3imtr4i 292 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅) |
13 | 8, 12 | ja 186 | . 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜑) → 𝐴 ≠ ∅) |
14 | 2, 13 | impbii 209 | 1 ⊢ (𝐴 ≠ ∅ ↔ (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∃wex 1776 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ∃wrex 3068 ∅c0 4339 |
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 |
This theorem depends on definitions: df-bi 207 df-an 396 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-dif 3966 df-nul 4340 |
This theorem is referenced by: iinpreima 7089 utopbas 24260 clsk3nimkb 44030 radcnvrat 44310 |
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