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Mirrors > Home > MPE Home > Th. List > Mathboxes > fineqvacALT | Structured version Visualization version GIF version |
Description: Shorter proof of fineqvac 33762 using ax-rep 5246 and ax-pow 5324. (Contributed by BTernaryTau, 21-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
fineqvacALT | ⊢ (Fin = V → CHOICE) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssv 3972 | . . . 4 ⊢ dom card ⊆ V | |
2 | 1 | a1i 11 | . . 3 ⊢ (Fin = V → dom card ⊆ V) |
3 | finnum 9892 | . . . . 5 ⊢ (𝑥 ∈ Fin → 𝑥 ∈ dom card) | |
4 | 3 | ssriv 3952 | . . . 4 ⊢ Fin ⊆ dom card |
5 | sseq1 3973 | . . . 4 ⊢ (Fin = V → (Fin ⊆ dom card ↔ V ⊆ dom card)) | |
6 | 4, 5 | mpbii 232 | . . 3 ⊢ (Fin = V → V ⊆ dom card) |
7 | 2, 6 | eqssd 3965 | . 2 ⊢ (Fin = V → dom card = V) |
8 | dfac10 10081 | . 2 ⊢ (CHOICE ↔ dom card = V) | |
9 | 7, 8 | sylibr 233 | 1 ⊢ (Fin = V → CHOICE) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 Vcvv 3447 ⊆ wss 3914 dom cdm 5637 Fincfn 8889 cardccrd 9879 CHOICEwac 10059 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-om 7807 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-er 8654 df-en 8890 df-fin 8893 df-card 9883 df-ac 10060 |
This theorem is referenced by: (None) |
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