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Mirrors > Home > NFE Home > Th. List > clos1nrel | GIF version |
Description: The value of a closure when the base set is not related to anything in R. (Contributed by SF, 13-Mar-2015.) |
Ref | Expression |
---|---|
clos1nrel.1 | ⊢ S ∈ V |
clos1nrel.2 | ⊢ R ∈ V |
clos1nrel.3 | ⊢ C = Clos1 (S, R) |
Ref | Expression |
---|---|
clos1nrel | ⊢ ((R “ S) = ∅ → C = S) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eq0 3564 | . . . . 5 ⊢ ((R “ S) = ∅ ↔ ∀y ¬ y ∈ (R “ S)) | |
2 | rspe 2675 | . . . . . . . . 9 ⊢ ((x ∈ S ∧ xRy) → ∃x ∈ S xRy) | |
3 | elima 4754 | . . . . . . . . 9 ⊢ (y ∈ (R “ S) ↔ ∃x ∈ S xRy) | |
4 | 2, 3 | sylibr 203 | . . . . . . . 8 ⊢ ((x ∈ S ∧ xRy) → y ∈ (R “ S)) |
5 | 4 | con3i 127 | . . . . . . 7 ⊢ (¬ y ∈ (R “ S) → ¬ (x ∈ S ∧ xRy)) |
6 | 5 | pm2.21d 98 | . . . . . 6 ⊢ (¬ y ∈ (R “ S) → ((x ∈ S ∧ xRy) → y ∈ S)) |
7 | 6 | alimi 1559 | . . . . 5 ⊢ (∀y ¬ y ∈ (R “ S) → ∀y((x ∈ S ∧ xRy) → y ∈ S)) |
8 | 1, 7 | sylbi 187 | . . . 4 ⊢ ((R “ S) = ∅ → ∀y((x ∈ S ∧ xRy) → y ∈ S)) |
9 | 8 | ralrimivw 2698 | . . 3 ⊢ ((R “ S) = ∅ → ∀x ∈ C ∀y((x ∈ S ∧ xRy) → y ∈ S)) |
10 | clos1nrel.1 | . . . 4 ⊢ S ∈ V | |
11 | ssid 3290 | . . . 4 ⊢ S ⊆ S | |
12 | clos1nrel.2 | . . . . 5 ⊢ R ∈ V | |
13 | clos1nrel.3 | . . . . 5 ⊢ C = Clos1 (S, R) | |
14 | 10, 12, 13 | clos1induct 5880 | . . . 4 ⊢ ((S ∈ V ∧ S ⊆ S ∧ ∀x ∈ C ∀y((x ∈ S ∧ xRy) → y ∈ S)) → C ⊆ S) |
15 | 10, 11, 14 | mp3an12 1267 | . . 3 ⊢ (∀x ∈ C ∀y((x ∈ S ∧ xRy) → y ∈ S) → C ⊆ S) |
16 | 9, 15 | syl 15 | . 2 ⊢ ((R “ S) = ∅ → C ⊆ S) |
17 | 13 | clos1base 5878 | . . 3 ⊢ S ⊆ C |
18 | 17 | a1i 10 | . 2 ⊢ ((R “ S) = ∅ → S ⊆ C) |
19 | 16, 18 | eqssd 3289 | 1 ⊢ ((R “ S) = ∅ → C = S) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 358 ∀wal 1540 = wceq 1642 ∈ wcel 1710 ∀wral 2614 ∃wrex 2615 Vcvv 2859 ⊆ wss 3257 ∅c0 3550 class class class wbr 4639 “ cima 4722 Clos1 cclos1 5872 |
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-sset 4725 df-co 4726 df-ima 4727 df-si 4728 df-id 4767 df-xp 4784 df-cnv 4785 df-rn 4786 df-dm 4787 df-res 4788 df-2nd 4797 df-txp 5736 df-fix 5740 df-ins2 5750 df-ins3 5752 df-image 5754 df-clos1 5873 |
This theorem is referenced by: nchoicelem3 6291 |
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