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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 7721. (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 5300 | . . . 4 ⊢ ∅ ∈ V | |
4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → ∅ ∈ V) |
5 | 1oex 8474 | . . . 4 ⊢ 1o ∈ V | |
6 | 5 | a1i 11 | . . 3 ⊢ (𝜑 → 1o ∈ V) |
7 | enpr2d.3 | . . . 4 ⊢ (𝜑 → ¬ 𝐴 = 𝐵) | |
8 | 7 | neqned 2941 | . . 3 ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
9 | 1n0 8486 | . . . . 5 ⊢ 1o ≠ ∅ | |
10 | 9 | necomi 2989 | . . . 4 ⊢ ∅ ≠ 1o |
11 | 10 | a1i 11 | . . 3 ⊢ (𝜑 → ∅ ≠ 1o) |
12 | 1, 2, 4, 6, 8, 11 | en2prd 9047 | . 2 ⊢ (𝜑 → {𝐴, 𝐵} ≈ {∅, 1o}) |
13 | df2o3 8472 | . 2 ⊢ 2o = {∅, 1o} | |
14 | 12, 13 | breqtrrdi 5183 | 1 ⊢ (𝜑 → {𝐴, 𝐵} ≈ 2o) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 Vcvv 3468 ∅c0 4317 {cpr 4625 class class class wbr 5141 1oc1o 8457 2oc2o 8458 ≈ cen 8935 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-suc 6363 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-1o 8464 df-2o 8465 df-en 8939 |
This theorem is referenced by: 1sdom2dom 9246 enpr2 9996 simpgnsgd 20020 2nsgsimpgd 20022 |
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