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Mirrors > Home > MPE Home > Th. List > Mathboxes > fineqvacALT | Structured version Visualization version GIF version |
Description: Shorter proof of fineqvac 32649 using ax-rep 5164 and ax-pow 5242. (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 3911 | . . . 4 ⊢ dom card ⊆ V | |
2 | 1 | a1i 11 | . . 3 ⊢ (Fin = V → dom card ⊆ V) |
3 | finnum 9462 | . . . . 5 ⊢ (𝑥 ∈ Fin → 𝑥 ∈ dom card) | |
4 | 3 | ssriv 3891 | . . . 4 ⊢ Fin ⊆ dom card |
5 | sseq1 3912 | . . . 4 ⊢ (Fin = V → (Fin ⊆ dom card ↔ V ⊆ dom card)) | |
6 | 4, 5 | mpbii 236 | . . 3 ⊢ (Fin = V → V ⊆ dom card) |
7 | 2, 6 | eqssd 3904 | . 2 ⊢ (Fin = V → dom card = V) |
8 | dfac10 9649 | . 2 ⊢ (CHOICE ↔ dom card = V) | |
9 | 7, 8 | sylibr 237 | 1 ⊢ (Fin = V → CHOICE) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 Vcvv 3400 ⊆ wss 3853 dom cdm 5535 Fincfn 8567 cardccrd 9449 CHOICEwac 9627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-int 4847 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-se 5494 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7139 df-om 7612 df-wrecs 7988 df-recs 8049 df-er 8332 df-en 8568 df-fin 8571 df-card 9453 df-ac 9628 |
This theorem is referenced by: (None) |
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