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Mirrors > Home > NFE Home > Th. List > releqel | GIF version |
Description: Lemma to turn a membership condition into an equality condition. (Contributed by SF, 9-Mar-2015.) |
Ref | Expression |
---|---|
releqel.1 | ⊢ T ∈ V |
releqel.2 | ⊢ (〈{y}, T〉 ∈ R ↔ y ∈ A) |
Ref | Expression |
---|---|
releqel | ⊢ (〈x, T〉 ∈ ∼ (( Ins3 S ⊕ Ins2 R) “ 1c) ↔ x = A) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elima1c 4947 | . . . 4 ⊢ (〈x, T〉 ∈ (( Ins3 S ⊕ Ins2 R) “ 1c) ↔ ∃y〈{y}, 〈x, T〉〉 ∈ ( Ins3 S ⊕ Ins2 R)) | |
2 | elsymdif 3223 | . . . . . 6 ⊢ (〈{y}, 〈x, T〉〉 ∈ ( Ins3 S ⊕ Ins2 R) ↔ ¬ (〈{y}, 〈x, T〉〉 ∈ Ins3 S ↔ 〈{y}, 〈x, T〉〉 ∈ Ins2 R)) | |
3 | releqel.1 | . . . . . . . . 9 ⊢ T ∈ V | |
4 | 3 | otelins3 5792 | . . . . . . . 8 ⊢ (〈{y}, 〈x, T〉〉 ∈ Ins3 S ↔ 〈{y}, x〉 ∈ S ) |
5 | vex 2862 | . . . . . . . . 9 ⊢ y ∈ V | |
6 | vex 2862 | . . . . . . . . 9 ⊢ x ∈ V | |
7 | 5, 6 | opelssetsn 4760 | . . . . . . . 8 ⊢ (〈{y}, x〉 ∈ S ↔ y ∈ x) |
8 | 4, 7 | bitri 240 | . . . . . . 7 ⊢ (〈{y}, 〈x, T〉〉 ∈ Ins3 S ↔ y ∈ x) |
9 | 6 | otelins2 5791 | . . . . . . . 8 ⊢ (〈{y}, 〈x, T〉〉 ∈ Ins2 R ↔ 〈{y}, T〉 ∈ R) |
10 | releqel.2 | . . . . . . . 8 ⊢ (〈{y}, T〉 ∈ R ↔ y ∈ A) | |
11 | 9, 10 | bitri 240 | . . . . . . 7 ⊢ (〈{y}, 〈x, T〉〉 ∈ Ins2 R ↔ y ∈ A) |
12 | 8, 11 | bibi12i 306 | . . . . . 6 ⊢ ((〈{y}, 〈x, T〉〉 ∈ Ins3 S ↔ 〈{y}, 〈x, T〉〉 ∈ Ins2 R) ↔ (y ∈ x ↔ y ∈ A)) |
13 | 2, 12 | xchbinx 301 | . . . . 5 ⊢ (〈{y}, 〈x, T〉〉 ∈ ( Ins3 S ⊕ Ins2 R) ↔ ¬ (y ∈ x ↔ y ∈ A)) |
14 | 13 | exbii 1582 | . . . 4 ⊢ (∃y〈{y}, 〈x, T〉〉 ∈ ( Ins3 S ⊕ Ins2 R) ↔ ∃y ¬ (y ∈ x ↔ y ∈ A)) |
15 | exnal 1574 | . . . 4 ⊢ (∃y ¬ (y ∈ x ↔ y ∈ A) ↔ ¬ ∀y(y ∈ x ↔ y ∈ A)) | |
16 | 1, 14, 15 | 3bitrri 263 | . . 3 ⊢ (¬ ∀y(y ∈ x ↔ y ∈ A) ↔ 〈x, T〉 ∈ (( Ins3 S ⊕ Ins2 R) “ 1c)) |
17 | 16 | con1bii 321 | . 2 ⊢ (¬ 〈x, T〉 ∈ (( Ins3 S ⊕ Ins2 R) “ 1c) ↔ ∀y(y ∈ x ↔ y ∈ A)) |
18 | 6, 3 | opex 4588 | . . 3 ⊢ 〈x, T〉 ∈ V |
19 | 18 | elcompl 3225 | . 2 ⊢ (〈x, T〉 ∈ ∼ (( Ins3 S ⊕ Ins2 R) “ 1c) ↔ ¬ 〈x, T〉 ∈ (( Ins3 S ⊕ Ins2 R) “ 1c)) |
20 | dfcleq 2347 | . 2 ⊢ (x = A ↔ ∀y(y ∈ x ↔ y ∈ A)) | |
21 | 17, 19, 20 | 3bitr4i 268 | 1 ⊢ (〈x, T〉 ∈ ∼ (( Ins3 S ⊕ Ins2 R) “ 1c) ↔ x = A) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 176 ∀wal 1540 ∃wex 1541 = wceq 1642 ∈ wcel 1710 Vcvv 2859 ∼ ccompl 3205 ⊕ csymdif 3209 {csn 3737 1cc1c 4134 〈cop 4561 S csset 4719 “ cima 4722 Ins2 cins2 5749 Ins3 cins3 5751 |
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-sset 4725 df-co 4726 df-ima 4727 df-cnv 4785 df-2nd 4797 df-txp 5736 df-ins2 5750 df-ins3 5752 |
This theorem is referenced by: releqmpt 5808 ceex 6174 nmembers1lem1 6268 nchoicelem10 6298 |
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