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Mirrors > Home > NFE Home > Th. List > erth | GIF version |
Description: Basic property of equivalence relations. Theorem 73 of [Suppes] p. 82. (Contributed by set.mm contributors, 23-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.) |
Ref | Expression |
---|---|
erth.1 | ⊢ (φ → R Er V) |
erth.2 | ⊢ (φ → dom R = X) |
erth.3 | ⊢ (φ → A ∈ X) |
erth.4 | ⊢ (φ → B ∈ V) |
Ref | Expression |
---|---|
erth | ⊢ (φ → (ARB ↔ [A]R = [B]R)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | erth.1 | . . . . . . . 8 ⊢ (φ → R Er V) | |
2 | 1 | adantr 451 | . . . . . . 7 ⊢ ((φ ∧ (ARB ∧ ARx)) → R Er V) |
3 | erth.4 | . . . . . . . . 9 ⊢ (φ → B ∈ V) | |
4 | elex 2867 | . . . . . . . . 9 ⊢ (B ∈ V → B ∈ V) | |
5 | 3, 4 | syl 15 | . . . . . . . 8 ⊢ (φ → B ∈ V) |
6 | 5 | adantr 451 | . . . . . . 7 ⊢ ((φ ∧ (ARB ∧ ARx)) → B ∈ V) |
7 | erth.3 | . . . . . . . . 9 ⊢ (φ → A ∈ X) | |
8 | elex 2867 | . . . . . . . . 9 ⊢ (A ∈ X → A ∈ V) | |
9 | 7, 8 | syl 15 | . . . . . . . 8 ⊢ (φ → A ∈ V) |
10 | 9 | adantr 451 | . . . . . . 7 ⊢ ((φ ∧ (ARB ∧ ARx)) → A ∈ V) |
11 | vex 2862 | . . . . . . . 8 ⊢ x ∈ V | |
12 | 11 | a1i 10 | . . . . . . 7 ⊢ ((φ ∧ (ARB ∧ ARx)) → x ∈ V) |
13 | simprl 732 | . . . . . . 7 ⊢ ((φ ∧ (ARB ∧ ARx)) → ARB) | |
14 | simprr 733 | . . . . . . 7 ⊢ ((φ ∧ (ARB ∧ ARx)) → ARx) | |
15 | 2, 6, 10, 12, 13, 14 | ertr3d 5957 | . . . . . 6 ⊢ ((φ ∧ (ARB ∧ ARx)) → BRx) |
16 | 15 | expr 598 | . . . . 5 ⊢ ((φ ∧ ARB) → (ARx → BRx)) |
17 | 11 | a1i 10 | . . . . . . 7 ⊢ (φ → x ∈ V) |
18 | 1, 9, 5, 17 | ertr 5954 | . . . . . 6 ⊢ (φ → ((ARB ∧ BRx) → ARx)) |
19 | 18 | expdimp 426 | . . . . 5 ⊢ ((φ ∧ ARB) → (BRx → ARx)) |
20 | 16, 19 | impbid 183 | . . . 4 ⊢ ((φ ∧ ARB) → (ARx ↔ BRx)) |
21 | 20 | abbidv 2467 | . . 3 ⊢ ((φ ∧ ARB) → {x ∣ ARx} = {x ∣ BRx}) |
22 | dfec2 5948 | . . 3 ⊢ [A]R = {x ∣ ARx} | |
23 | dfec2 5948 | . . 3 ⊢ [B]R = {x ∣ BRx} | |
24 | 21, 22, 23 | 3eqtr4g 2410 | . 2 ⊢ ((φ ∧ ARB) → [A]R = [B]R) |
25 | 1 | adantr 451 | . . 3 ⊢ ((φ ∧ [A]R = [B]R) → R Er V) |
26 | simpl 443 | . . . 4 ⊢ ((φ ∧ [A]R = [B]R) → φ) | |
27 | 26, 3, 4 | 3syl 18 | . . 3 ⊢ ((φ ∧ [A]R = [B]R) → B ∈ V) |
28 | 26, 7, 8 | 3syl 18 | . . 3 ⊢ ((φ ∧ [A]R = [B]R) → A ∈ V) |
29 | erth.2 | . . . . . . . 8 ⊢ (φ → dom R = X) | |
30 | 1, 29, 7 | erref 5959 | . . . . . . 7 ⊢ (φ → ARA) |
31 | elec 5964 | . . . . . . 7 ⊢ (A ∈ [A]R ↔ ARA) | |
32 | 30, 31 | sylibr 203 | . . . . . 6 ⊢ (φ → A ∈ [A]R) |
33 | eleq2 2414 | . . . . . 6 ⊢ ([A]R = [B]R → (A ∈ [A]R ↔ A ∈ [B]R)) | |
34 | 32, 33 | syl5ibcom 211 | . . . . 5 ⊢ (φ → ([A]R = [B]R → A ∈ [B]R)) |
35 | 34 | imp 418 | . . . 4 ⊢ ((φ ∧ [A]R = [B]R) → A ∈ [B]R) |
36 | elec 5964 | . . . 4 ⊢ (A ∈ [B]R ↔ BRA) | |
37 | 35, 36 | sylib 188 | . . 3 ⊢ ((φ ∧ [A]R = [B]R) → BRA) |
38 | 25, 27, 28, 37 | ersym 5952 | . 2 ⊢ ((φ ∧ [A]R = [B]R) → ARB) |
39 | 24, 38 | impbida 805 | 1 ⊢ (φ → (ARB ↔ [A]R = [B]R)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 = wceq 1642 ∈ wcel 1710 {cab 2339 Vcvv 2859 class class class wbr 4639 dom cdm 4772 Er cer 5898 [cec 5945 |
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-ima 4727 df-xp 4784 df-cnv 4785 df-rn 4786 df-dm 4787 df-res 4788 df-trans 5899 df-sym 5908 df-er 5909 df-ec 5947 |
This theorem is referenced by: erth2 5969 erthi 5970 eqncg 6126 ncseqnc 6128 |
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