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Mirrors > Home > NFE Home > Th. List > erref | GIF version |
Description: An equivalence relation is reflexive on its field. Compare Theorem 3M of [Enderton] p. 56. (Contributed by set.mm contributors, 6-May-2013.) |
Ref | Expression |
---|---|
erref.1 | ⊢ (φ → R Er V) |
erref.2 | ⊢ (φ → dom R = A) |
erref.3 | ⊢ (φ → X ∈ A) |
Ref | Expression |
---|---|
erref | ⊢ (φ → XRX) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | erref.3 | . 2 ⊢ (φ → X ∈ A) | |
2 | erref.2 | . . . 4 ⊢ (φ → dom R = A) | |
3 | 2 | eleq2d 2420 | . . 3 ⊢ (φ → (X ∈ dom R ↔ X ∈ A)) |
4 | eldm 4899 | . . . 4 ⊢ (X ∈ dom R ↔ ∃y XRy) | |
5 | erref.1 | . . . . . . . 8 ⊢ (φ → R Er V) | |
6 | 5 | adantr 451 | . . . . . . 7 ⊢ ((φ ∧ XRy) → R Er V) |
7 | elex 2868 | . . . . . . . . 9 ⊢ (X ∈ A → X ∈ V) | |
8 | 1, 7 | syl 15 | . . . . . . . 8 ⊢ (φ → X ∈ V) |
9 | 8 | adantr 451 | . . . . . . 7 ⊢ ((φ ∧ XRy) → X ∈ V) |
10 | vex 2863 | . . . . . . . 8 ⊢ y ∈ V | |
11 | 10 | a1i 10 | . . . . . . 7 ⊢ ((φ ∧ XRy) → y ∈ V) |
12 | simpr 447 | . . . . . . 7 ⊢ ((φ ∧ XRy) → XRy) | |
13 | 6, 9, 11, 9, 12, 12 | ertr4d 5959 | . . . . . 6 ⊢ ((φ ∧ XRy) → XRX) |
14 | 13 | ex 423 | . . . . 5 ⊢ (φ → (XRy → XRX)) |
15 | 14 | exlimdv 1636 | . . . 4 ⊢ (φ → (∃y XRy → XRX)) |
16 | 4, 15 | syl5bi 208 | . . 3 ⊢ (φ → (X ∈ dom R → XRX)) |
17 | 3, 16 | sylbird 226 | . 2 ⊢ (φ → (X ∈ A → XRX)) |
18 | 1, 17 | mpd 14 | 1 ⊢ (φ → XRX) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 ∃wex 1541 = wceq 1642 ∈ wcel 1710 Vcvv 2860 class class class wbr 4640 dom cdm 4773 Er cer 5899 |
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 4079 ax-xp 4080 ax-cnv 4081 ax-1c 4082 ax-sset 4083 ax-si 4084 ax-ins2 4085 ax-ins3 4086 ax-typlower 4087 ax-sn 4088 |
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 2479 df-ne 2519 df-ral 2620 df-rex 2621 df-reu 2622 df-rmo 2623 df-rab 2624 df-v 2862 df-sbc 3048 df-nin 3212 df-compl 3213 df-in 3214 df-un 3215 df-dif 3216 df-symdif 3217 df-ss 3260 df-pss 3262 df-nul 3552 df-if 3664 df-pw 3725 df-sn 3742 df-pr 3743 df-uni 3893 df-int 3928 df-opk 4059 df-1c 4137 df-pw1 4138 df-uni1 4139 df-xpk 4186 df-cnvk 4187 df-ins2k 4188 df-ins3k 4189 df-imak 4190 df-cok 4191 df-p6 4192 df-sik 4193 df-ssetk 4194 df-imagek 4195 df-idk 4196 df-iota 4340 df-0c 4378 df-addc 4379 df-nnc 4380 df-fin 4381 df-lefin 4441 df-ltfin 4442 df-ncfin 4443 df-tfin 4444 df-evenfin 4445 df-oddfin 4446 df-sfin 4447 df-spfin 4448 df-phi 4566 df-op 4567 df-proj1 4568 df-proj2 4569 df-opab 4624 df-br 4641 df-ima 4728 df-cnv 4786 df-rn 4787 df-dm 4788 df-trans 5900 df-sym 5909 df-er 5910 |
This theorem is referenced by: erth 5969 |
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