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Mirrors > Home > MPE Home > Th. List > Mathboxes > snen1g | Structured version Visualization version GIF version |
Description: A singleton is equinumerous to ordinal one iff its content is a set. (Contributed by RP, 8-Oct-2023.) |
Ref | Expression |
---|---|
snen1g | ⊢ ({𝐴} ≈ 1o ↔ 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqcom 2828 | . . . 4 ⊢ ({𝐴} = {𝑥} ↔ {𝑥} = {𝐴}) | |
2 | vex 3497 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | 2 | sneqr 4771 | . . . . 5 ⊢ ({𝑥} = {𝐴} → 𝑥 = 𝐴) |
4 | sneq 4577 | . . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
5 | 3, 4 | impbii 211 | . . . 4 ⊢ ({𝑥} = {𝐴} ↔ 𝑥 = 𝐴) |
6 | 1, 5 | bitri 277 | . . 3 ⊢ ({𝐴} = {𝑥} ↔ 𝑥 = 𝐴) |
7 | 6 | exbii 1848 | . 2 ⊢ (∃𝑥{𝐴} = {𝑥} ↔ ∃𝑥 𝑥 = 𝐴) |
8 | en1 8576 | . 2 ⊢ ({𝐴} ≈ 1o ↔ ∃𝑥{𝐴} = {𝑥}) | |
9 | isset 3506 | . 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
10 | 7, 8, 9 | 3bitr4i 305 | 1 ⊢ ({𝐴} ≈ 1o ↔ 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∃wex 1780 ∈ wcel 2114 Vcvv 3494 {csn 4567 class class class wbr 5066 1oc1o 8095 ≈ cen 8506 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-1o 8102 df-en 8510 |
This theorem is referenced by: snen1el 39940 |
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