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Mirrors > Home > NFE Home > Th. List > elxp2 | GIF version |
Description: Membership in a cross product. (Contributed by NM, 23-Feb-2004.) |
Ref | Expression |
---|---|
elxp2 | ⊢ (A ∈ (B × C) ↔ ∃x ∈ B ∃y ∈ C A = 〈x, y〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rex 2620 | . . . 4 ⊢ (∃y ∈ C (x ∈ B ∧ A = 〈x, y〉) ↔ ∃y(y ∈ C ∧ (x ∈ B ∧ A = 〈x, y〉))) | |
2 | r19.42v 2765 | . . . 4 ⊢ (∃y ∈ C (x ∈ B ∧ A = 〈x, y〉) ↔ (x ∈ B ∧ ∃y ∈ C A = 〈x, y〉)) | |
3 | an13 774 | . . . . 5 ⊢ ((y ∈ C ∧ (x ∈ B ∧ A = 〈x, y〉)) ↔ (A = 〈x, y〉 ∧ (x ∈ B ∧ y ∈ C))) | |
4 | 3 | exbii 1582 | . . . 4 ⊢ (∃y(y ∈ C ∧ (x ∈ B ∧ A = 〈x, y〉)) ↔ ∃y(A = 〈x, y〉 ∧ (x ∈ B ∧ y ∈ C))) |
5 | 1, 2, 4 | 3bitr3i 266 | . . 3 ⊢ ((x ∈ B ∧ ∃y ∈ C A = 〈x, y〉) ↔ ∃y(A = 〈x, y〉 ∧ (x ∈ B ∧ y ∈ C))) |
6 | 5 | exbii 1582 | . 2 ⊢ (∃x(x ∈ B ∧ ∃y ∈ C A = 〈x, y〉) ↔ ∃x∃y(A = 〈x, y〉 ∧ (x ∈ B ∧ y ∈ C))) |
7 | df-rex 2620 | . 2 ⊢ (∃x ∈ B ∃y ∈ C A = 〈x, y〉 ↔ ∃x(x ∈ B ∧ ∃y ∈ C A = 〈x, y〉)) | |
8 | elxp 4801 | . 2 ⊢ (A ∈ (B × C) ↔ ∃x∃y(A = 〈x, y〉 ∧ (x ∈ B ∧ y ∈ C))) | |
9 | 6, 7, 8 | 3bitr4ri 269 | 1 ⊢ (A ∈ (B × C) ↔ ∃x ∈ B ∃y ∈ C A = 〈x, y〉) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∧ wa 358 ∃wex 1541 = wceq 1642 ∈ wcel 1710 ∃wrex 2615 〈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-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-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 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-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-addc 4378 df-nnc 4379 df-phi 4565 df-op 4566 df-opab 4623 df-xp 4784 |
This theorem is referenced by: xpiundi 4817 xpiundir 4818 dfxp2 5113 xpnedisj 5513 1st2nd2 5516 crossex 5850 xpassen 6057 addccan2nclem1 6263 |
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