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Mirrors > Home > MPE Home > Th. List > Mathboxes > c0exALT | Structured version Visualization version GIF version |
Description: Alternate proof of c0ex 11204 using more set theory axioms but fewer complex number axioms (add ax-10 2137, ax-11 2154, ax-13 2371, ax-nul 5305, and remove ax-1cn 11164, ax-icn 11165, ax-addcl 11166, and ax-mulcl 11168). (Contributed by Steven Nguyen, 4-Dec-2022.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
c0exALT | ⊢ 0 ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-i2m1 11174 | . . 3 ⊢ ((i · i) + 1) = 0 | |
2 | 1 | eqcomi 2741 | . 2 ⊢ 0 = ((i · i) + 1) |
3 | 2 | ovexi 7439 | 1 ⊢ 0 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 Vcvv 3474 (class class class)co 7405 0cc0 11106 1c1 11107 ici 11108 + caddc 11109 · cmul 11111 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-nul 5305 ax-i2m1 11174 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-sn 4628 df-pr 4630 df-uni 4908 df-iota 6492 df-fv 6548 df-ov 7408 |
This theorem is referenced by: (None) |
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