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Mirrors > Home > NFE Home > Th. List > connexd | GIF version |
Description: The connectivity property. (Contributed by SF, 18-Mar-2015.) |
Ref | Expression |
---|---|
connexd.1 | ⊢ (φ → R Connex A) |
connexd.2 | ⊢ (φ → X ∈ A) |
connexd.3 | ⊢ (φ → Y ∈ A) |
Ref | Expression |
---|---|
connexd | ⊢ (φ → (XRY ∨ YRX)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | connexd.1 | . 2 ⊢ (φ → R Connex A) | |
2 | brex 4689 | . . . . 5 ⊢ (R Connex A → (R ∈ V ∧ A ∈ V)) | |
3 | breq 4641 | . . . . . . . 8 ⊢ (r = R → (xry ↔ xRy)) | |
4 | breq 4641 | . . . . . . . 8 ⊢ (r = R → (yrx ↔ yRx)) | |
5 | 3, 4 | orbi12d 690 | . . . . . . 7 ⊢ (r = R → ((xry ∨ yrx) ↔ (xRy ∨ yRx))) |
6 | 5 | 2ralbidv 2656 | . . . . . 6 ⊢ (r = R → (∀x ∈ a ∀y ∈ a (xry ∨ yrx) ↔ ∀x ∈ a ∀y ∈ a (xRy ∨ yRx))) |
7 | raleq 2807 | . . . . . . 7 ⊢ (a = A → (∀y ∈ a (xRy ∨ yRx) ↔ ∀y ∈ A (xRy ∨ yRx))) | |
8 | 7 | raleqbi1dv 2815 | . . . . . 6 ⊢ (a = A → (∀x ∈ a ∀y ∈ a (xRy ∨ yRx) ↔ ∀x ∈ A ∀y ∈ A (xRy ∨ yRx))) |
9 | df-connex 5903 | . . . . . 6 ⊢ Connex = {〈r, a〉 ∣ ∀x ∈ a ∀y ∈ a (xry ∨ yrx)} | |
10 | 6, 8, 9 | brabg 4706 | . . . . 5 ⊢ ((R ∈ V ∧ A ∈ V) → (R Connex A ↔ ∀x ∈ A ∀y ∈ A (xRy ∨ yRx))) |
11 | 2, 10 | syl 15 | . . . 4 ⊢ (R Connex A → (R Connex A ↔ ∀x ∈ A ∀y ∈ A (xRy ∨ yRx))) |
12 | 11 | ibi 232 | . . 3 ⊢ (R Connex A → ∀x ∈ A ∀y ∈ A (xRy ∨ yRx)) |
13 | connexd.2 | . . . 4 ⊢ (φ → X ∈ A) | |
14 | connexd.3 | . . . 4 ⊢ (φ → Y ∈ A) | |
15 | breq1 4642 | . . . . . 6 ⊢ (x = X → (xRy ↔ XRy)) | |
16 | breq2 4643 | . . . . . 6 ⊢ (x = X → (yRx ↔ yRX)) | |
17 | 15, 16 | orbi12d 690 | . . . . 5 ⊢ (x = X → ((xRy ∨ yRx) ↔ (XRy ∨ yRX))) |
18 | breq2 4643 | . . . . . 6 ⊢ (y = Y → (XRy ↔ XRY)) | |
19 | breq1 4642 | . . . . . 6 ⊢ (y = Y → (yRX ↔ YRX)) | |
20 | 18, 19 | orbi12d 690 | . . . . 5 ⊢ (y = Y → ((XRy ∨ yRX) ↔ (XRY ∨ YRX))) |
21 | 17, 20 | rspc2v 2961 | . . . 4 ⊢ ((X ∈ A ∧ Y ∈ A) → (∀x ∈ A ∀y ∈ A (xRy ∨ yRx) → (XRY ∨ YRX))) |
22 | 13, 14, 21 | syl2anc 642 | . . 3 ⊢ (φ → (∀x ∈ A ∀y ∈ A (xRy ∨ yRx) → (XRY ∨ YRX))) |
23 | 12, 22 | syl5 28 | . 2 ⊢ (φ → (R Connex A → (XRY ∨ YRX))) |
24 | 1, 23 | mpd 14 | 1 ⊢ (φ → (XRY ∨ YRX)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∨ wo 357 ∧ wa 358 = wceq 1642 ∈ wcel 1710 ∀wral 2614 Vcvv 2859 class class class wbr 4639 Connex cconnex 5892 |
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-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-connex 5903 |
This theorem is referenced by: weds 5938 nchoicelem8 6296 |
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