New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > xpnz | GIF version |
Description: The cross product of nonempty classes is nonempty. (Variation of a theorem contributed by Raph Levien, 30-Jun-2006.) (Contributed by set.mm contributors, 30-Jun-2006.) (Revised by set.mm contributors, 19-Apr-2007.) |
Ref | Expression |
---|---|
xpnz | ⊢ ((A ≠ ∅ ∧ B ≠ ∅) ↔ (A × B) ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0 3559 | . . . . 5 ⊢ (A ≠ ∅ ↔ ∃x x ∈ A) | |
2 | n0 3559 | . . . . 5 ⊢ (B ≠ ∅ ↔ ∃y y ∈ B) | |
3 | 1, 2 | anbi12i 678 | . . . 4 ⊢ ((A ≠ ∅ ∧ B ≠ ∅) ↔ (∃x x ∈ A ∧ ∃y y ∈ B)) |
4 | eeanv 1913 | . . . 4 ⊢ (∃x∃y(x ∈ A ∧ y ∈ B) ↔ (∃x x ∈ A ∧ ∃y y ∈ B)) | |
5 | 3, 4 | bitr4i 243 | . . 3 ⊢ ((A ≠ ∅ ∧ B ≠ ∅) ↔ ∃x∃y(x ∈ A ∧ y ∈ B)) |
6 | opelxp 4811 | . . . . 5 ⊢ (〈x, y〉 ∈ (A × B) ↔ (x ∈ A ∧ y ∈ B)) | |
7 | ne0i 3556 | . . . . 5 ⊢ (〈x, y〉 ∈ (A × B) → (A × B) ≠ ∅) | |
8 | 6, 7 | sylbir 204 | . . . 4 ⊢ ((x ∈ A ∧ y ∈ B) → (A × B) ≠ ∅) |
9 | 8 | exlimivv 1635 | . . 3 ⊢ (∃x∃y(x ∈ A ∧ y ∈ B) → (A × B) ≠ ∅) |
10 | 5, 9 | sylbi 187 | . 2 ⊢ ((A ≠ ∅ ∧ B ≠ ∅) → (A × B) ≠ ∅) |
11 | xpeq1 4798 | . . . . 5 ⊢ (A = ∅ → (A × B) = (∅ × B)) | |
12 | xp0r 4842 | . . . . 5 ⊢ (∅ × B) = ∅ | |
13 | 11, 12 | syl6eq 2401 | . . . 4 ⊢ (A = ∅ → (A × B) = ∅) |
14 | 13 | necon3i 2555 | . . 3 ⊢ ((A × B) ≠ ∅ → A ≠ ∅) |
15 | xpeq2 4799 | . . . . 5 ⊢ (B = ∅ → (A × B) = (A × ∅)) | |
16 | xp0 5044 | . . . . 5 ⊢ (A × ∅) = ∅ | |
17 | 15, 16 | syl6eq 2401 | . . . 4 ⊢ (B = ∅ → (A × B) = ∅) |
18 | 17 | necon3i 2555 | . . 3 ⊢ ((A × B) ≠ ∅ → B ≠ ∅) |
19 | 14, 18 | jca 518 | . 2 ⊢ ((A × B) ≠ ∅ → (A ≠ ∅ ∧ B ≠ ∅)) |
20 | 10, 19 | impbii 180 | 1 ⊢ ((A ≠ ∅ ∧ B ≠ ∅) ↔ (A × B) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∧ wa 358 ∃wex 1541 = wceq 1642 ∈ wcel 1710 ≠ wne 2516 ∅c0 3550 〈cop 4561 × cxp 4770 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-xp 4784 df-cnv 4785 |
This theorem is referenced by: xpeq0 5046 ssxpb 5055 xp11 5056 xpexr2 5110 |
Copyright terms: Public domain | W3C validator |