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Mirrors > Home > NFE Home > Th. List > xpsnen | GIF version |
Description: A set is equinumerous to its cross-product with a singleton. Proposition 4.22(c) of [Mendelson] p. 254. (Contributed by set.mm contributors, 23-Feb-2015.) |
Ref | Expression |
---|---|
xpsnen.1 | ⊢ A ∈ V |
xpsnen.2 | ⊢ B ∈ V |
Ref | Expression |
---|---|
xpsnen | ⊢ (A × {B}) ≈ A |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpsnen.2 | . . . . 5 ⊢ B ∈ V | |
2 | 1 | snid 3760 | . . . 4 ⊢ B ∈ {B} |
3 | ne0i 3556 | . . . 4 ⊢ (B ∈ {B} → {B} ≠ ∅) | |
4 | dmxp 4923 | . . . 4 ⊢ ({B} ≠ ∅ → dom (A × {B}) = A) | |
5 | 2, 3, 4 | mp2b 9 | . . 3 ⊢ dom (A × {B}) = A |
6 | 1 | fconst 5250 | . . . 4 ⊢ (A × {B}):A–→{B} |
7 | ffun 5225 | . . . 4 ⊢ ((A × {B}):A–→{B} → Fun (A × {B})) | |
8 | xpsnen.1 | . . . . . 6 ⊢ A ∈ V | |
9 | snex 4111 | . . . . . 6 ⊢ {B} ∈ V | |
10 | 8, 9 | xpex 5115 | . . . . 5 ⊢ (A × {B}) ∈ V |
11 | 10 | fundmen 6043 | . . . 4 ⊢ (Fun (A × {B}) → dom (A × {B}) ≈ (A × {B})) |
12 | 6, 7, 11 | mp2b 9 | . . 3 ⊢ dom (A × {B}) ≈ (A × {B}) |
13 | 5, 12 | eqbrtrri 4660 | . 2 ⊢ A ≈ (A × {B}) |
14 | ensym 6037 | . 2 ⊢ (A ≈ (A × {B}) ↔ (A × {B}) ≈ A) | |
15 | 13, 14 | mpbi 199 | 1 ⊢ (A × {B}) ≈ A |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1642 ∈ wcel 1710 ≠ wne 2516 Vcvv 2859 ∅c0 3550 {csn 3737 class class class wbr 4639 × cxp 4770 dom cdm 4772 Fun wfun 4775 –→wf 4777 ≈ cen 6028 |
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-1st 4723 df-swap 4724 df-co 4726 df-ima 4727 df-id 4767 df-xp 4784 df-cnv 4785 df-rn 4786 df-dm 4787 df-res 4788 df-fun 4789 df-fn 4790 df-f 4791 df-f1 4792 df-fo 4793 df-f1o 4794 df-2nd 4797 df-en 6029 |
This theorem is referenced by: xpsneng 6050 endisj 6051 mucid1 6143 ncaddccl 6144 tcdi 6164 ce0addcnnul 6179 |
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