New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > clos1basesucg | GIF version |
Description: A member of a closure is either in the base set or connected to another member by R. Theorem IX.5.16 of [Rosser] p. 248. (Contributed by Scott Fenton, 31-Jul-2019.) |
Ref | Expression |
---|---|
clos1basesucg.1 | ⊢ C = Clos1 (S, R) |
Ref | Expression |
---|---|
clos1basesucg | ⊢ ((S ∈ V ∧ R ∈ W) → (A ∈ C ↔ (A ∈ S ∨ ∃x ∈ C xRA))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clos1eq1 5874 | . . . . 5 ⊢ (s = S → Clos1 (s, r) = Clos1 (S, r)) | |
2 | 1 | eleq2d 2420 | . . . 4 ⊢ (s = S → (A ∈ Clos1 (s, r) ↔ A ∈ Clos1 (S, r))) |
3 | eleq2 2414 | . . . . 5 ⊢ (s = S → (A ∈ s ↔ A ∈ S)) | |
4 | 1 | rexeqdv 2814 | . . . . 5 ⊢ (s = S → (∃x ∈ Clos1 (s, r)xrA ↔ ∃x ∈ Clos1 (S, r)xrA)) |
5 | 3, 4 | orbi12d 690 | . . . 4 ⊢ (s = S → ((A ∈ s ∨ ∃x ∈ Clos1 (s, r)xrA) ↔ (A ∈ S ∨ ∃x ∈ Clos1 (S, r)xrA))) |
6 | 2, 5 | bibi12d 312 | . . 3 ⊢ (s = S → ((A ∈ Clos1 (s, r) ↔ (A ∈ s ∨ ∃x ∈ Clos1 (s, r)xrA)) ↔ (A ∈ Clos1 (S, r) ↔ (A ∈ S ∨ ∃x ∈ Clos1 (S, r)xrA)))) |
7 | clos1eq2 5875 | . . . . 5 ⊢ (r = R → Clos1 (S, r) = Clos1 (S, R)) | |
8 | 7 | eleq2d 2420 | . . . 4 ⊢ (r = R → (A ∈ Clos1 (S, r) ↔ A ∈ Clos1 (S, R))) |
9 | breq 4641 | . . . . . 6 ⊢ (r = R → (xrA ↔ xRA)) | |
10 | 7, 9 | rexeqbidv 2820 | . . . . 5 ⊢ (r = R → (∃x ∈ Clos1 (S, r)xrA ↔ ∃x ∈ Clos1 (S, R)xRA)) |
11 | 10 | orbi2d 682 | . . . 4 ⊢ (r = R → ((A ∈ S ∨ ∃x ∈ Clos1 (S, r)xrA) ↔ (A ∈ S ∨ ∃x ∈ Clos1 (S, R)xRA))) |
12 | 8, 11 | bibi12d 312 | . . 3 ⊢ (r = R → ((A ∈ Clos1 (S, r) ↔ (A ∈ S ∨ ∃x ∈ Clos1 (S, r)xrA)) ↔ (A ∈ Clos1 (S, R) ↔ (A ∈ S ∨ ∃x ∈ Clos1 (S, R)xRA)))) |
13 | vex 2862 | . . . 4 ⊢ s ∈ V | |
14 | vex 2862 | . . . 4 ⊢ r ∈ V | |
15 | eqid 2353 | . . . 4 ⊢ Clos1 (s, r) = Clos1 (s, r) | |
16 | 13, 14, 15 | clos1basesuc 5882 | . . 3 ⊢ (A ∈ Clos1 (s, r) ↔ (A ∈ s ∨ ∃x ∈ Clos1 (s, r)xrA)) |
17 | 6, 12, 16 | vtocl2g 2918 | . 2 ⊢ ((S ∈ V ∧ R ∈ W) → (A ∈ Clos1 (S, R) ↔ (A ∈ S ∨ ∃x ∈ Clos1 (S, R)xRA))) |
18 | clos1basesucg.1 | . . 3 ⊢ C = Clos1 (S, R) | |
19 | 18 | eleq2i 2417 | . 2 ⊢ (A ∈ C ↔ A ∈ Clos1 (S, R)) |
20 | 18 | rexeqi 2812 | . . 3 ⊢ (∃x ∈ C xRA ↔ ∃x ∈ Clos1 (S, R)xRA) |
21 | 20 | orbi2i 505 | . 2 ⊢ ((A ∈ S ∨ ∃x ∈ C xRA) ↔ (A ∈ S ∨ ∃x ∈ Clos1 (S, R)xRA)) |
22 | 17, 19, 21 | 3bitr4g 279 | 1 ⊢ ((S ∈ V ∧ R ∈ W) → (A ∈ C ↔ (A ∈ S ∨ ∃x ∈ C xRA))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∨ wo 357 ∧ wa 358 = wceq 1642 ∈ wcel 1710 ∃wrex 2615 class class class wbr 4639 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: dmfrec 6316 fnfreclem2 6318 fnfreclem3 6319 frec0 6321 frecsuc 6322 |
Copyright terms: Public domain | W3C validator |