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Mirrors > Home > MPE Home > Th. List > snmapen1 | Structured version Visualization version GIF version |
Description: Set exponentiation: a singleton to any set is equinumerous to ordinal 1. (Proposed by BJ, 17-Jul-2022.) (Contributed by AV, 17-Jul-2022.) |
Ref | Expression |
---|---|
snmapen1 | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} ↑𝑚 𝐵) ≈ 1o) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snmapen 8302 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} ↑𝑚 𝐵) ≈ {𝐴}) | |
2 | ensn1g 8286 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ 1o) | |
3 | 2 | adantr 474 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴} ≈ 1o) |
4 | entr 8273 | . 2 ⊢ ((({𝐴} ↑𝑚 𝐵) ≈ {𝐴} ∧ {𝐴} ≈ 1o) → ({𝐴} ↑𝑚 𝐵) ≈ 1o) | |
5 | 1, 3, 4 | syl2anc 581 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} ↑𝑚 𝐵) ≈ 1o) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2166 {csn 4396 class class class wbr 4872 (class class class)co 6904 1oc1o 7818 ↑𝑚 cmap 8121 ≈ cen 8218 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-ral 3121 df-rex 3122 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-op 4403 df-uni 4658 df-br 4873 df-opab 4935 df-mpt 4952 df-id 5249 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-suc 5968 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-1o 7825 df-er 8008 df-map 8123 df-en 8222 |
This theorem is referenced by: map1 8304 |
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