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Mirrors > Home > MPE Home > Th. List > Mathboxes > snen1el | Structured version Visualization version GIF version |
Description: A singleton is equinumerous to ordinal one if its content is an element of it. (Contributed by RP, 8-Oct-2023.) |
Ref | Expression |
---|---|
snen1el | ⊢ ({𝐴} ≈ 1o ↔ 𝐴 ∈ {𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snen1g 39965 | . 2 ⊢ ({𝐴} ≈ 1o ↔ 𝐴 ∈ V) | |
2 | snidb 4593 | . 2 ⊢ (𝐴 ∈ V ↔ 𝐴 ∈ {𝐴}) | |
3 | 1, 2 | bitri 277 | 1 ⊢ ({𝐴} ≈ 1o ↔ 𝐴 ∈ {𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∈ wcel 2113 Vcvv 3491 {csn 4560 class class class wbr 5059 1oc1o 8088 ≈ cen 8499 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-1o 8095 df-en 8503 |
This theorem is referenced by: (None) |
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