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Mirrors > Home > NFE Home > Th. List > xpexr | GIF version |
Description: If a cross product is a set, one of its components must be a set. (Contributed by set.mm contributors, 27-Aug-2006.) |
Ref | Expression |
---|---|
xpexr | ⊢ ((A × B) ∈ C → (A ∈ V ∨ B ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 4110 | . . . . . 6 ⊢ ∅ ∈ V | |
2 | eleq1 2413 | . . . . . 6 ⊢ (A = ∅ → (A ∈ V ↔ ∅ ∈ V)) | |
3 | 1, 2 | mpbiri 224 | . . . . 5 ⊢ (A = ∅ → A ∈ V) |
4 | 3 | pm2.24d 135 | . . . 4 ⊢ (A = ∅ → (¬ A ∈ V → B ∈ V)) |
5 | 4 | a1d 22 | . . 3 ⊢ (A = ∅ → ((A × B) ∈ C → (¬ A ∈ V → B ∈ V))) |
6 | rnexg 5104 | . . . . 5 ⊢ ((A × B) ∈ C → ran (A × B) ∈ V) | |
7 | rnxp 5051 | . . . . . 6 ⊢ (A ≠ ∅ → ran (A × B) = B) | |
8 | 7 | eleq1d 2419 | . . . . 5 ⊢ (A ≠ ∅ → (ran (A × B) ∈ V ↔ B ∈ V)) |
9 | 6, 8 | syl5ib 210 | . . . 4 ⊢ (A ≠ ∅ → ((A × B) ∈ C → B ∈ V)) |
10 | 9 | a1dd 42 | . . 3 ⊢ (A ≠ ∅ → ((A × B) ∈ C → (¬ A ∈ V → B ∈ V))) |
11 | 5, 10 | pm2.61ine 2592 | . 2 ⊢ ((A × B) ∈ C → (¬ A ∈ V → B ∈ V)) |
12 | 11 | orrd 367 | 1 ⊢ ((A × B) ∈ C → (A ∈ V ∨ B ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 357 = wceq 1642 ∈ wcel 1710 ≠ wne 2516 Vcvv 2859 ∅c0 3550 × cxp 4770 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-ima 4727 df-xp 4784 df-cnv 4785 df-rn 4786 df-dm 4787 |
This theorem is referenced by: (None) |
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