Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ixp0 | Structured version Visualization version GIF version |
Description: The infinite Cartesian product of a family 𝐵(𝑥) with an empty member is empty. The converse of this theorem is equivalent to the Axiom of Choice, see ac9 9305. (Contributed by NM, 1-Oct-2006.) (Proof shortened by Mario Carneiro, 22-Jun-2016.) |
Ref | Expression |
---|---|
ixp0 | ⊢ (∃𝑥 ∈ 𝐴 𝐵 = ∅ → X𝑥 ∈ 𝐴 𝐵 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nne 2798 | . . . 4 ⊢ (¬ 𝐵 ≠ ∅ ↔ 𝐵 = ∅) | |
2 | 1 | rexbii 3041 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝐵 ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝐵 = ∅) |
3 | rexnal 2995 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝐵 ≠ ∅ ↔ ¬ ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅) | |
4 | 2, 3 | bitr3i 266 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝐵 = ∅ ↔ ¬ ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅) |
5 | ixpn0 7940 | . . 3 ⊢ (X𝑥 ∈ 𝐴 𝐵 ≠ ∅ → ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅) | |
6 | 5 | necon1bi 2822 | . 2 ⊢ (¬ ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅ → X𝑥 ∈ 𝐴 𝐵 = ∅) |
7 | 4, 6 | sylbi 207 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝐵 = ∅ → X𝑥 ∈ 𝐴 𝐵 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1483 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 ∅c0 3915 Xcixp 7908 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-v 3202 df-dif 3577 df-nul 3916 df-ixp 7909 |
This theorem is referenced by: vonioo 40896 vonicc 40899 |
Copyright terms: Public domain | W3C validator |