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Mirrors > Home > MPE Home > Th. List > en1b | Structured version Visualization version GIF version |
Description: A set is equinumerous to ordinal one iff it is a singleton. (Contributed by Mario Carneiro, 17-Jan-2015.) Avoid ax-un 7754. (Revised by BTernaryTau, 24-Sep-2024.) |
Ref | Expression |
---|---|
en1b | ⊢ (𝐴 ≈ 1o ↔ 𝐴 = {∪ 𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | en1 9063 | . . 3 ⊢ (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥}) | |
2 | id 22 | . . . . 5 ⊢ (𝐴 = {𝑥} → 𝐴 = {𝑥}) | |
3 | unieq 4923 | . . . . . . 7 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = ∪ {𝑥}) | |
4 | unisnv 4932 | . . . . . . 7 ⊢ ∪ {𝑥} = 𝑥 | |
5 | 3, 4 | eqtrdi 2791 | . . . . . 6 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = 𝑥) |
6 | 5 | sneqd 4643 | . . . . 5 ⊢ (𝐴 = {𝑥} → {∪ 𝐴} = {𝑥}) |
7 | 2, 6 | eqtr4d 2778 | . . . 4 ⊢ (𝐴 = {𝑥} → 𝐴 = {∪ 𝐴}) |
8 | 7 | exlimiv 1928 | . . 3 ⊢ (∃𝑥 𝐴 = {𝑥} → 𝐴 = {∪ 𝐴}) |
9 | 1, 8 | sylbi 217 | . 2 ⊢ (𝐴 ≈ 1o → 𝐴 = {∪ 𝐴}) |
10 | id 22 | . . 3 ⊢ (𝐴 = {∪ 𝐴} → 𝐴 = {∪ 𝐴}) | |
11 | eqsnuniex 5367 | . . . 4 ⊢ (𝐴 = {∪ 𝐴} → ∪ 𝐴 ∈ V) | |
12 | ensn1g 9061 | . . . 4 ⊢ (∪ 𝐴 ∈ V → {∪ 𝐴} ≈ 1o) | |
13 | 11, 12 | syl 17 | . . 3 ⊢ (𝐴 = {∪ 𝐴} → {∪ 𝐴} ≈ 1o) |
14 | 10, 13 | eqbrtrd 5170 | . 2 ⊢ (𝐴 = {∪ 𝐴} → 𝐴 ≈ 1o) |
15 | 9, 14 | impbii 209 | 1 ⊢ (𝐴 ≈ 1o ↔ 𝐴 = {∪ 𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1537 ∃wex 1776 ∈ wcel 2106 Vcvv 3478 {csn 4631 ∪ cuni 4912 class class class wbr 5148 1oc1o 8498 ≈ cen 8981 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-1o 8505 df-en 8985 |
This theorem is referenced by: en1uniel 9068 sylow2alem2 19651 sylow2a 19652 frgpcyg 21610 ptcmplem3 24078 cnextfvval 24089 cnextcn 24091 minveclem4a 25478 isppw 27172 xrge0tsmsbi 33049 |
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