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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 4398. 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 4398 | . . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝜑) → ∃𝑥 ∈ 𝐴 𝜑) | |
2 | 1 | ex 416 | . 2 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜑)) |
3 | noel 4247 | . . . . . . 7 ⊢ ¬ 𝑥 ∈ ∅ | |
4 | 3 | pm2.21i 119 | . . . . . 6 ⊢ (𝑥 ∈ ∅ → 𝜑) |
5 | 4 | rgen 3116 | . . . . 5 ⊢ ∀𝑥 ∈ ∅ 𝜑 |
6 | raleq 3358 | . . . . 5 ⊢ (𝐴 = ∅ → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ ∅ 𝜑)) | |
7 | 5, 6 | mpbiri 261 | . . . 4 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑) |
8 | 7 | necon3bi 3013 | . . 3 ⊢ (¬ ∀𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅) |
9 | exsimpl 1869 | . . . 4 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥 𝑥 ∈ 𝐴) | |
10 | df-rex 3112 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
11 | n0 4260 | . . . 4 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
12 | 9, 10, 11 | 3imtr4i 295 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅) |
13 | 8, 12 | ja 189 | . 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜑) → 𝐴 ≠ ∅) |
14 | 2, 13 | impbii 212 | 1 ⊢ (𝐴 ≠ ∅ ↔ (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ∃wrex 3107 ∅c0 4243 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ne 2988 df-ral 3111 df-rex 3112 df-dif 3884 df-nul 4244 |
This theorem is referenced by: iinpreima 6814 utopbas 22841 clsk3nimkb 40743 radcnvrat 41018 |
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