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| Mirrors > Home > MPE Home > Th. List > Mathboxes > c0exALT | Structured version Visualization version GIF version | ||
| Description: Alternate proof of c0ex 11255 using more set theory axioms but fewer complex number axioms (add ax-10 2141, ax-11 2157, ax-13 2377, ax-nul 5306, and remove ax-1cn 11213, ax-icn 11214, ax-addcl 11215, and ax-mulcl 11217). (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 11223 | . . 3 ⊢ ((i · i) + 1) = 0 | |
| 2 | 1 | eqcomi 2746 | . 2 ⊢ 0 = ((i · i) + 1) |
| 3 | 2 | ovexi 7465 | 1 ⊢ 0 ∈ V |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 Vcvv 3480 (class class class)co 7431 0cc0 11155 1c1 11156 ici 11157 + caddc 11158 · cmul 11160 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 ax-i2m1 11223 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-sn 4627 df-pr 4629 df-uni 4908 df-iota 6514 df-fv 6569 df-ov 7434 |
| This theorem is referenced by: (None) |
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