![]() |
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 7740. (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 5307 | . . . 4 ⊢ ∅ ∈ V | |
4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → ∅ ∈ V) |
5 | 1oex 8497 | . . . 4 ⊢ 1o ∈ V | |
6 | 5 | a1i 11 | . . 3 ⊢ (𝜑 → 1o ∈ V) |
7 | enpr2d.3 | . . . 4 ⊢ (𝜑 → ¬ 𝐴 = 𝐵) | |
8 | 7 | neqned 2944 | . . 3 ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
9 | 1n0 8509 | . . . . 5 ⊢ 1o ≠ ∅ | |
10 | 9 | necomi 2992 | . . . 4 ⊢ ∅ ≠ 1o |
11 | 10 | a1i 11 | . . 3 ⊢ (𝜑 → ∅ ≠ 1o) |
12 | 1, 2, 4, 6, 8, 11 | en2prd 9073 | . 2 ⊢ (𝜑 → {𝐴, 𝐵} ≈ {∅, 1o}) |
13 | df2o3 8495 | . 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 1534 ∈ wcel 2099 ≠ wne 2937 Vcvv 3471 ∅c0 4323 {cpr 4631 class class class wbr 5148 1oc1o 8480 2oc2o 8481 ≈ cen 8961 |
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-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-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-opab 5211 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-suc 6375 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-1o 8487 df-2o 8488 df-en 8965 |
This theorem is referenced by: 1sdom2dom 9272 enpr2 10026 simpgnsgd 20057 2nsgsimpgd 20059 |
Copyright terms: Public domain | W3C validator |