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Mirrors > Home > MPE Home > Th. List > grpon0 | Structured version Visualization version GIF version |
Description: The base set of a group is not empty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
grpfo.1 | ⊢ 𝑋 = ran 𝐺 |
Ref | Expression |
---|---|
grpon0 | ⊢ (𝐺 ∈ GrpOp → 𝑋 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpfo.1 | . . 3 ⊢ 𝑋 = ran 𝐺 | |
2 | 1 | grpolidinv 28536 | . 2 ⊢ (𝐺 ∈ GrpOp → ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢)) |
3 | rexn0 4408 | . 2 ⊢ (∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢) → 𝑋 ≠ ∅) | |
4 | 2, 3 | syl 17 | 1 ⊢ (𝐺 ∈ GrpOp → 𝑋 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 ∀wral 3051 ∃wrex 3052 ∅c0 4223 ran crn 5537 (class class class)co 7191 GrpOpcgr 28524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-fo 6364 df-fv 6366 df-ov 7194 df-grpo 28528 |
This theorem is referenced by: 0ngrp 28546 rngone0 35755 |
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