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Mirrors > Home > MPE Home > Th. List > tc0 | Structured version Visualization version GIF version |
Description: The transitive closure of the empty set. (Contributed by Mario Carneiro, 4-Jun-2015.) |
Ref | Expression |
---|---|
tc0 | ⊢ (TC‘∅) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3872 | . . 3 ⊢ ∅ ⊆ ∅ | |
2 | tr0 5037 | . . 3 ⊢ Tr ∅ | |
3 | 0ex 5064 | . . . 4 ⊢ ∅ ∈ V | |
4 | tcmin 8975 | . . . 4 ⊢ (∅ ∈ V → ((∅ ⊆ ∅ ∧ Tr ∅) → (TC‘∅) ⊆ ∅)) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ ((∅ ⊆ ∅ ∧ Tr ∅) → (TC‘∅) ⊆ ∅) |
6 | 1, 2, 5 | mp2an 680 | . 2 ⊢ (TC‘∅) ⊆ ∅ |
7 | 0ss 4230 | . 2 ⊢ ∅ ⊆ (TC‘∅) | |
8 | 6, 7 | eqssi 3867 | 1 ⊢ (TC‘∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1508 ∈ wcel 2051 Vcvv 3408 ⊆ wss 3822 ∅c0 4172 Tr wtr 5026 ‘cfv 6185 TCctc 8970 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-inf2 8896 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-ral 3086 df-rex 3087 df-reu 3088 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-om 7395 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-tc 8971 |
This theorem is referenced by: tc00 8982 |
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