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Mirrors > Home > MPE Home > Th. List > Mathboxes > en2pr | Structured version Visualization version GIF version |
Description: A class is equinumerous to ordinal two iff it is a pair of distinct sets. (Contributed by RP, 11-Oct-2023.) |
Ref | Expression |
---|---|
en2pr | ⊢ (𝐴 ≈ 2o ↔ ∃𝑥∃𝑦(𝐴 = {𝑥, 𝑦} ∧ 𝑥 ≠ 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | en2 9305 | . . 3 ⊢ (𝐴 ≈ 2o → ∃𝑥∃𝑦 𝐴 = {𝑥, 𝑦}) | |
2 | 1 | pm4.71ri 560 | . 2 ⊢ (𝐴 ≈ 2o ↔ (∃𝑥∃𝑦 𝐴 = {𝑥, 𝑦} ∧ 𝐴 ≈ 2o)) |
3 | 19.41vv 1947 | . 2 ⊢ (∃𝑥∃𝑦(𝐴 = {𝑥, 𝑦} ∧ 𝐴 ≈ 2o) ↔ (∃𝑥∃𝑦 𝐴 = {𝑥, 𝑦} ∧ 𝐴 ≈ 2o)) | |
4 | breq1 5151 | . . . . 5 ⊢ (𝐴 = {𝑥, 𝑦} → (𝐴 ≈ 2o ↔ {𝑥, 𝑦} ≈ 2o)) | |
5 | pr2ne 10027 | . . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → ({𝑥, 𝑦} ≈ 2o ↔ 𝑥 ≠ 𝑦)) | |
6 | 5 | el2v 3479 | . . . . 5 ⊢ ({𝑥, 𝑦} ≈ 2o ↔ 𝑥 ≠ 𝑦) |
7 | 4, 6 | bitrdi 287 | . . . 4 ⊢ (𝐴 = {𝑥, 𝑦} → (𝐴 ≈ 2o ↔ 𝑥 ≠ 𝑦)) |
8 | 7 | pm5.32i 574 | . . 3 ⊢ ((𝐴 = {𝑥, 𝑦} ∧ 𝐴 ≈ 2o) ↔ (𝐴 = {𝑥, 𝑦} ∧ 𝑥 ≠ 𝑦)) |
9 | 8 | 2exbii 1844 | . 2 ⊢ (∃𝑥∃𝑦(𝐴 = {𝑥, 𝑦} ∧ 𝐴 ≈ 2o) ↔ ∃𝑥∃𝑦(𝐴 = {𝑥, 𝑦} ∧ 𝑥 ≠ 𝑦)) |
10 | 2, 3, 9 | 3bitr2i 299 | 1 ⊢ (𝐴 ≈ 2o ↔ ∃𝑥∃𝑦(𝐴 = {𝑥, 𝑦} ∧ 𝑥 ≠ 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1534 ∃wex 1774 ≠ wne 2937 Vcvv 3471 {cpr 4631 class class class wbr 5148 2oc2o 8480 ≈ cen 8960 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ord 6372 df-on 6373 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-1o 8486 df-2o 8487 df-en 8964 |
This theorem is referenced by: (None) |
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