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Mirrors > Home > MPE Home > Th. List > Mathboxes > c0exALT | Structured version Visualization version GIF version |
Description: Alternate proof of c0ex 10629 using more set theory axioms but fewer complex number axioms (add ax-10 2141, ax-11 2157, ax-13 2386, ax-nul 5202, and remove ax-1cn 10589, ax-icn 10590, ax-addcl 10591, and ax-mulcl 10593). (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 10599 | . . 3 ⊢ ((i · i) + 1) = 0 | |
2 | 1 | eqcomi 2830 | . 2 ⊢ 0 = ((i · i) + 1) |
3 | 2 | ovexi 7184 | 1 ⊢ 0 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2110 Vcvv 3494 (class class class)co 7150 0cc0 10531 1c1 10532 ici 10533 + caddc 10534 · cmul 10536 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-nul 5202 ax-i2m1 10599 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-sn 4561 df-pr 4563 df-uni 4832 df-iota 6308 df-fv 6357 df-ov 7153 |
This theorem is referenced by: (None) |
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