![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > enpr2d | Structured version Visualization version GIF version |
Description: A pair with distinct elements is equinumerous to ordinal two. (Contributed by Rohan Ridenour, 3-Aug-2023.) Avoid ax-un 7754. (Revised by BTernaryTau, 23-Dec-2024.) |
Ref | Expression |
---|---|
enpr2d.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
enpr2d.2 | ⊢ (𝜑 → 𝐵 ∈ 𝐷) |
enpr2d.3 | ⊢ (𝜑 → ¬ 𝐴 = 𝐵) |
Ref | Expression |
---|---|
enpr2d | ⊢ (𝜑 → {𝐴, 𝐵} ≈ 2o) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | enpr2d.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐶) | |
2 | enpr2d.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝐷) | |
3 | 0ex 5313 | . . . 4 ⊢ ∅ ∈ V | |
4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → ∅ ∈ V) |
5 | 1oex 8515 | . . . 4 ⊢ 1o ∈ V | |
6 | 5 | a1i 11 | . . 3 ⊢ (𝜑 → 1o ∈ V) |
7 | enpr2d.3 | . . . 4 ⊢ (𝜑 → ¬ 𝐴 = 𝐵) | |
8 | 7 | neqned 2945 | . . 3 ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
9 | 1n0 8525 | . . . . 5 ⊢ 1o ≠ ∅ | |
10 | 9 | necomi 2993 | . . . 4 ⊢ ∅ ≠ 1o |
11 | 10 | a1i 11 | . . 3 ⊢ (𝜑 → ∅ ≠ 1o) |
12 | 1, 2, 4, 6, 8, 11 | en2prd 9087 | . 2 ⊢ (𝜑 → {𝐴, 𝐵} ≈ {∅, 1o}) |
13 | df2o3 8513 | . 2 ⊢ 2o = {∅, 1o} | |
14 | 12, 13 | breqtrrdi 5190 | 1 ⊢ (𝜑 → {𝐴, 𝐵} ≈ 2o) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 Vcvv 3478 ∅c0 4339 {cpr 4633 class class class wbr 5148 1oc1o 8498 2oc2o 8499 ≈ 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-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-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 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-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-suc 6392 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-1o 8505 df-2o 8506 df-en 8985 |
This theorem is referenced by: 1sdom2dom 9281 prfi 9361 enpr2 10040 simpgnsgd 20135 2nsgsimpgd 20137 |
Copyright terms: Public domain | W3C validator |