New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > xpexr2 | GIF version |
Description: If a nonempty cross product is a set, so are both of its components. (Contributed by set.mm contributors, 27-Aug-2006.) (Revised by set.mm contributors, 5-May-2007.) |
Ref | Expression |
---|---|
xpexr2 | ⊢ (((A × B) ∈ C ∧ (A × B) ≠ ∅) → (A ∈ V ∧ B ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpnz 5045 | . 2 ⊢ ((A ≠ ∅ ∧ B ≠ ∅) ↔ (A × B) ≠ ∅) | |
2 | dmxp 4923 | . . . . . 6 ⊢ (B ≠ ∅ → dom (A × B) = A) | |
3 | 2 | adantl 452 | . . . . 5 ⊢ (((A × B) ∈ C ∧ B ≠ ∅) → dom (A × B) = A) |
4 | dmexg 5105 | . . . . . 6 ⊢ ((A × B) ∈ C → dom (A × B) ∈ V) | |
5 | 4 | adantr 451 | . . . . 5 ⊢ (((A × B) ∈ C ∧ B ≠ ∅) → dom (A × B) ∈ V) |
6 | 3, 5 | eqeltrrd 2428 | . . . 4 ⊢ (((A × B) ∈ C ∧ B ≠ ∅) → A ∈ V) |
7 | rnxp 5051 | . . . . . 6 ⊢ (A ≠ ∅ → ran (A × B) = B) | |
8 | 7 | adantl 452 | . . . . 5 ⊢ (((A × B) ∈ C ∧ A ≠ ∅) → ran (A × B) = B) |
9 | rnexg 5104 | . . . . . 6 ⊢ ((A × B) ∈ C → ran (A × B) ∈ V) | |
10 | 9 | adantr 451 | . . . . 5 ⊢ (((A × B) ∈ C ∧ A ≠ ∅) → ran (A × B) ∈ V) |
11 | 8, 10 | eqeltrrd 2428 | . . . 4 ⊢ (((A × B) ∈ C ∧ A ≠ ∅) → B ∈ V) |
12 | 6, 11 | anim12dan 810 | . . 3 ⊢ (((A × B) ∈ C ∧ (B ≠ ∅ ∧ A ≠ ∅)) → (A ∈ V ∧ B ∈ V)) |
13 | 12 | ancom2s 777 | . 2 ⊢ (((A × B) ∈ C ∧ (A ≠ ∅ ∧ B ≠ ∅)) → (A ∈ V ∧ B ∈ V)) |
14 | 1, 13 | sylan2br 462 | 1 ⊢ (((A × B) ∈ C ∧ (A × B) ≠ ∅) → (A ∈ V ∧ B ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 = wceq 1642 ∈ wcel 1710 ≠ wne 2516 Vcvv 2859 ∅c0 3550 × cxp 4770 dom cdm 4772 ran crn 4773 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 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-swap 4724 df-ima 4727 df-xp 4784 df-cnv 4785 df-rn 4786 df-dm 4787 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |